At every instant 'now' the spacetime
points are determined by matter, and
have dual structure: the spacetime it is
both irreversibly fixed in the past and
indefinable in the future. At every
instant 'now' points emerge, and have structure (FR = 1)
exhibited with Point I and Points II.

Every point is emerging --
one-at-a-time -- in the
Arrow of Space as dual
object: it is both irreversibly fixed in the past by
Points IIand "open" (indefinable by
matter) in Point I (global mode of spacetime). The
"separator" between Point I and Points II is the
instant 'now'. The Cauchy limit is
the final endpoint C (Point I)
from which Points II emerge in the irreversible past as
perfect continuum (called local mode of spacetime) in
which dt/ds is effectively non-existent, while at
the same instant 'now' the initial Point I offers the
next 'open set' of re-created Points II to
be chosen from the potential
future of the Arrow of Space in the
next elementary step dt/ds. Because every
point is dual object, it is also suggested that the logic
of propositions and truth statements must be YAIN (Yes
And neIN).

The Universe has indefinable boundary at
C .No function can be defined on the very endpoint
C (Point I). In the
Cauchy limit Ansatz [ε(..........)ε] ,
the untraceable endpoint
C is excluded by using open
intervals only. Surely with actual infinity we can think
like bartenders and obtain the
physical Points II (always in plural), but never
the endpoint C (Point I) itself.

Perfectly smooth torus-sphere transition via endpoint
C in the
so-called global mode of spacetime of Point I (the
Universe as ONE).

The small red
circle contains the
Dedekind cut
C in the infinite, unphysical, and
non-Archimedean spacetime (Point I) of the loop
'now' (see
below), obtained with actual
infinity.
An asymptotically flat spacetime (called
flash or slice) corresponds
to the local (physical) mode of spacetime. It is made of
physical PointsII which can be individuated with
matter (the Cheshire cat) and hence
obtain point-like numbers, included imprecise ones from
irrationals.

The four quadrants below are mirror imagesobtained by
replacing (t) with
(-t) and 'left' with
'right' (not shown). The atemporal loop 'now'
is nested within Point I (endpoint
C)in the
non-Archimedean global mode of spacetime.

Spacetime
quadrants in Relative Scale
gravity
(the favicon of this website is
inserted as decoration only)

Outline

In Relative Scale (RS) gravity, the
emergence of asymptotic boundaries of spacetime in
the Large (B) and the emergence of physical points in
the Small (B) are produced en blocby Point I ,
with Points II. We shall introduce Point I: a non-Archimedean, uncountably infinite,
purely
geometrical (a grin without the Cheshire cat),
and potential (yet-to-become physicalized) entity inhabiting the so-called global
mode of spacetime from which the Cauchy limit
and Dedekind Schnitt (C)
are projected in the local (physical) mode of spacetime by Points
II -- one-at-a-time
along the Arrow of Space. In RS gravity, the
whole local
(physical) mode of spacetime (called also flash)
is being re-created
en bloc in two directions, toward the Large (B)
and the Small (B), starting from
A (multiplicative
identity) in null "directions".

This is the only possible solution to the
problems of set theory and Continuum
Hypothesis: the "carrier" acting withindt/ds has been set to zero (perfect
continuum) bythe "speed" of light,
hence producing an ever-increasing past
by Points II,
while at the same time the potential, yet-to-become physical
state of the universe is presented with uncountably infinite
(no metric can be defined on null surfaces)
and purely geometrical Point I (the grin without the cat)
residing in the indefinable non-Archimedean global mode of spacetime.

The Aristotelian Connection (AC)
along the w-axis of
the Arrow of Space

Taking the risk to be
terribly boring again, I will introduce an example for 'potential
reality' from General Relativity (GR): the
reference fluid and 'individuating
field'. For reasons which I haven't been able to understand in the
past 40 years, people frantically believe that GR were 'classical
theory'. But it isn't, because it can't. Surely GR is not
quantum theory, but is not classical theory either. In addition to
the arguments from Erich Kretschmann (Über den physikalischen Sinn
der Relativitätspostulate, Annalen der Physik53
(1917) 575-614), in GR "fixation of a frame of reference and gauge
transformations are intertwined in a manner not encountered in any
other area of physics" (Peter
Bergmann, 1988), which brings insurmountable problems to the
reference fluid and 'individuating
field'. As John Stachel explained in 1993 (pp.
139-140), "there is no structure on the differentiable manifold
that is both independent of the metric tensor and able
to serve as an individuating field", in order to uniquely
identity "the points of the manifold by some property or
properties that characterize(s) each of the points."

As
Clifford
Will et al. put it, "the principle of general covariance,
upon which general relativity is built, implies that coordinates are
simply labels of spacetime events that can be assigned completely
arbitrarily (subject to some conditions of smoothness and
differentiability). The only quantities that have physical meaning –
the measurables – are those that are invariant under coordinate
transformations. One such invariant is the number of ticks on an
atomic clock giving the proper time between two events."

The first two
sentences from the excerpt above are clear: an object will remain
'the same' if we look at it from different directions, just as a
house remains invariant under different coordinates from different
maps, say. These are invariants. But are they 'observables'?

NB:Not in GR,
ladies and gentlemen. The invariant objects in GR resemble
Platonic ideas, which are UNspeakable
and physicallyindefinable. If we say, for example, 'when
it rains it pours', we apply particular "coordinates" (words) to
express an entity that can be equally well expressed with many
different "coordinates" (languages), because it will always remain
an invariant object, called here 'potential reality'.
In GR, the same phenomenon is called 'reference
fluid' and 'individuating field', thanks to which we have an
exact 'one meter' and exact 'one second' as invariant
objects. Just like Platonic ideas, these
invariants cannot be directly observed -- we can physically
observe only their "shadows" cast with different "coordinates", and
of course require that "coordinates are simply labels".

But look at the last
sentence in the excerpt above: "One such invariant is the number of
ticks on an atomic clock giving the proper time between two events."
I strongly disagree: the phenomenon which creates time as
dt cannot be temporal. Same tallies to space.

We can only try to
reproduce these invariants in metrology, and inevitably
use a
finite number of physical constituents. We cannot
use physical Points II cast from the invariant
'one second' residing as 'potential reality' at Point I. The
claim that an atomic clock "gives" the proper time is tantamount to
saying that your morning coffee is hot because it contains many tiny
little and very hot "particles".

These invariants
produce the physical spacetime of Points II (local
mode of spacetime). In Relative Scale gravity,
we further postulate that these invariants are dual.
Namely, they "expand" toward the Small (B)
and "contract" toward the Large (B),
starting from
A in null
"directions", yet a co-moving observer will always observe
one
and the same 'meter', be it an electron or a galaxy; see
below.

To cut the long story
short, gravity does not produce "curvature".
It only "shrinks" the invariant 'one meter', after which
bodies moves by the
principle of least action, and hence are "attracted"
until they become neutralized by the opposite
centrifugal force: dynamical equilibrium. At scales larger than
our solar system we encounter gravitational
"dark" effect and further at Hubble scale its mirrored
effect, called "dark energy".

Whether you believe you can do a thing or believe you can't, you are right,
said
Henry Ford. Hence the only way to "predict"
the future is to create it;
see my first paper about biocausality from
January 1990.

The so-called Scale Relativity
Principle was announced on 21
September 2008; the main idea is to remove the background
structure in GR, introduced with 'metric of space', as explained on
25 January 2005.
Very briefly, the current model of 3-D space is based on absolute
relations such as 'inside vs. outside' and 'small vs. large volume
of space', which are not acceptable. We need to eliminate all
absolute notions pertaining to 'space', to match the absence of explicit "time parameter" in GR (Karel Kuchar),
and suggest a new metric theory of gravity, in which the spacetime
itself is an
emergent phenomenon
explicated
from a metaphysical pre-geometric quantum-gravitational "dough";
see the Kochen-Specker Theorem and
Wilson chamber.

To avoid confusion with
other theories, the terms have been changed to Relative Scale
Principle (RSP) viz. Relative Scale (RS) theory of gravity. It is a
theory of quantum gravity in which the "quantization" of spacetime
is introduced from the outset with two modes of spacetime,
local (physical) and global, and the "verdammten
Quantenspringerei" (Erwin
Schrödinger,
1926) are interpreted as artifacts of the macroscopic
measuring devices: Dead matter
makes quantum jumps; the living-and-quantum matter is
smarter.

According to RSP, the
geometry of spacetime is effectively Archimedean only at macroscopic length
scale, at the lowest part from the drawing above (fixed observer A), while in
the directions toward
the two ends (10-35 m and 1026 m) a special kind of
fusion (also known as 'mutual penetration of Small and Large') occurs,
relative to co-moving observers B.
Namely, a macroscopic table with length 1 m will be "shrunk"
or "expanded", relative to
a fixed observer A, along the
"opposite" directions toward the two ends, yet
the metric of spacetime will also change accordingly: relative to co-moving
observers B "travelling" toward the ends of the drawing, a table
with length 1 m will always keep its RS-size of 1 m, although
the same table will be observed by the fixed macroscopic observer
A as 'small like an electron' or 'large like a
galaxy'. That
is, relative to the fixed observer A at the length scale of tables
and chairs, the spacetime is indeed Archimedean,
while relative to the co-moving observers B the same spacetime
undergoes mutual fusion by keeping invariant length of all
objects toward the ends of the above drawing.

Who has 'the
right scale'? Nobody, according to RSP, because all contradictory
estimates of length, relative to observers A and B, are in
fact
correct -- the scale itself is dual. Namely, an
object at Hubble scale will indeed be "large" and an elementary
particle will be "small" to observer A and its Archimedean
geometry, while at the same time the "two" (in fact, one)
object(s) B will be entangled and will keep its invariant
RS-size 'one and the same' in their respective domains pertaining to
"two" (in fact, one) observer(s) B. In their respective domain(s)
'out there', an RS-large object does not contain many
RS-small objects: both a galaxy and an electron are made of one
and the same
uncountably infinite "number" of geometrical points (Georg
Cantor). Stated differently, the "two" RS-templates, cast in
opposite "directions" with respect only to observer A, are in
fact one entangled object with invariant RS "size". Hence the
universe itself does not change its size but its metric, and
evolves along the Arrow of Space as ONE bootstrapped
self-regulated entity endowed with self-acting faculty from
its
Aristotelian First Cause. The fact that we can
think
about our brain, by our brain, makes our brain
'self-acting'. Physically, we cannot observe our mind in
the brain -- just a self-acting brain. In the case of
'the universe as ONE brain', simply replace self-acting
brain with 'sufficient conditions for
spacetime'.

According to
Relative Scale (RS) theory of gravity, there is one and only one
"direction" which begins at the macroscopic world with Archimedean
geometry (fixed observer A), with two dual
presentations toward the Small and the Large,
cast in the local (physical) mode of spacetime. This dual
"direction" is denoted in the Arrow of Space
with w-axis,
and is being nullified by the "speed"
of light -- one-instant-now-at-a-time. There is no
background resembling some "canvas" (John
Baez), but a re-created "back bone" of the whole universe
at all
length scales, made by matter itself -- one-point-at-a-time.

The main applications
of RSP are to suggest two presentations of entanglement,
quantum and gravitational,
which do not occur in the spacetime of a fixed macroscopic observer
A
-- the underlying phenomenon of entanglement occurs in the fused spacetime with
non-Archimedean geometry toward the Small
and the Large, and the dual RS-distance controlled by entanglement remains
one and the same. And secondly, the fundamental object which
facilitates the entanglement is 'the universe as ONE',
as explained with Kochen-Specker Theorem and
Wilson chamber. Thus, the RS
theory of gravity explains the phenomenon of
curvature-and-rotation as alteration
of spacetime metric by 'the universe as ONE'
(cf. sufficient conditions for spacetime):
the so-called "dark matter" and "dark energy" are
interpreted as
tug-of-war and time-symmetric
presentations of RS gravity, corresponding to "shrinking" and
"expanding" of the metric with respect to a macroscopic
observer A, while the actual distance between all point in
the fused spacetime remains one and the same with
respect to observers B. Stated differently, the cases of
RS-shrunk or RS-expanded metric (viz. positive or negative
curvature-and-rotation) are explained without any localized
physical "dark matter" or "dark energy", just as in the case of 'spin'.

Bear in mind that the
current interpretation of gravity inevitably leads to
"the worst
theoretical prediction in the history of
physics!" (Wiki).
In particular, the radius of the universe “could not even reach to
the moon,” as calculated by
Wolfgang Pauli. To understand how we produced such staggering
reductio ad absurdum, recall the tacit idea 'only matter can
interact with matter'. Namely, if we observe "anomalous"
gravitational effects at length scales exceeding the size of our
Solar System, we claim that such effects can only be caused by
matter. Fine, but what matter, and how?

The
phenomenon of transience,
exhibited in the Heracliteanflow of
time (cf. option YAIN (iii) below), can
exist only and exclusively only in a
self-acting universe: it is produced by
'the universe as ONE' (cf. sufficient conditions
for spacetime), which acts on itself by virtue of the
Aristotelian
First Cause. In RS theory of gravity, this self-acting
action is called Aristotelian Connection, and is presented with an
infinitesimal "displacement"
dt in the
Arrow of Space. Its source (Luke
17:21) cannot be traced back from its physical effects, just
as we cannot detect the human mind in its brain -- physically, we
can only observe a self-acting brain. And because the
universe is modeled as 'ONE brain', its
self-action is facilitated by alteration of geometry and
topology of the whole universe, which affects the distribution and
dynamics of matter localized at allentangled
spacetime points, en bloc. Which is why the so-called "cold
dark matter" and "dark energy of (you-name-it)"
do not exist, just like the physical basis of 'spin'.
Such Machian effects may be caused by 'all matter in the whole
universe', but their quasi-local
implementation by 'the universe as ONE' is with
two purely geometrical cases of
curvature-and-rotation, such as RS-shrunk or
RS-expanded metric. We should just forget about
supermassive black holes, "280 million solar
masses per particle," etc., and focus our efforts on
Einstein's theory of 'total
field of as yet unknown structure'.

Last but not least, the alteration of spacetime metric,
according to RS theory of gravity, is an effortless
phenomenon, because it requires alteration of the phase of
gravitational waves. The energy release (e.g.,
1054 ergs/pulse inGRBs)
is produced by the "engine" of the universe -- the
Arrow of Space. Similar to the "quantum
waves", the gravitational waves do not possess
intrinsic energy and can be manipulated
effortlessly, just like we drive our thoughts in our brains. The same
effortless action is performed jointly by 'the universe as a
brain' and its complementary (Wolfgang Pauli) Universal Mind (Henry
Margenau). As Max Planck
stated in
1944, "All matter originates and exists only by virtue of a
force. We must assume behind this force the existence of a conscious
and intelligent Mind. This Mind is the matrix of all matter". Or
simply “God's
thoughts,”Albert Einstein.

Albert Einstein: The
present
formulation of General Relativity (GR) is "merely
a makeshift in order to give the general principle of relativity a
preliminary closed-form expression. For it was essentially no more
than a theory of the gravitational field, which was isolated
somewhat artificially from a total field of as yet unknown
structure."

The main unsolved
task in Einstein's unfinished GR is the presentation of matter
("timber"):

Firstly, the
density of matter in the energy-momentum tensor (Erik
Curiel; Babak and Grishchuk) is presented
with
some continuous "dough", ignoring its quantum
structure, which in turn
leads to
"the worst
theoretical prediction in the history of
physics!" (Wiki).
Secondly, in metric theory of gravity there is no physical
gravitational energy obeying conservation law (Jose
Geraldo Pereira): the conversion (Hans
Ohanian) of "marble" into
"timber" (Hermann Bondi)
must be presented in such way that only the "timber" can do
work, but not the "marble" itself. The latter must not
obey Newton's third law,
as the inertial mass of an accelerating
particle is not "a back-reaction to its own gravitational
field" (Wolfgang Rindler,
p. 22). And thirdly, theconversion between the "marble" and its
physicalized "timber" is a bi-directional
"talk" (cf. below), which makes matter ("timber") self-acting.
Why? Because the other party ("marble") is hidden
by the "speed" of light.

NB: As
anticipated by Albert Einstein,
the solution to "marble-timber" relations can indeed be derived "from
a total field of as yet unknown structure": it
could be a hypothetical General Platonic Idea (GPI) field of the
joint phenomenon 'entanglement & flow of time', under the
stipulation that 'the universe as ONE' functions
as a brain as well.
Physically, we cannot observe the mind inside its brain -- just a self-acting brain.

In the case of
'the universe as ONE brain', we replace 'self-acting
brain' with 'sufficient conditions for
spacetime', to recover a holistic quantum-gravitational
phenomenon which determines -- one-at-a-time -- the quasi-local
mass of quasi-local fish, bootstrapped (Geoffrey
Chew) by their school of fish. The end
result from
'entanglement & flow of time' is a wave pattern of the
"timber", like the wave-like holomovement of
centipede's legs. That's how we see "waves"
without any physical source that would otherwise have to
jitter or pulsate in space, due to energy loss.
(Forget about
dimensionless "strain" h.)

1.3. Ansatz:
Gravity is not quantum phenomenon, for the same reasons why quantum
entanglement is not producing "dark"
gravitational effects (rotation & curvature).
Instead, the underlying phenomenon is entanglement
& flow of time in Relative Scale (RS)
gravity.

2.
Entanglement & flow of time: Arrow of Space

2.1. The emergence
of spacetime (Isham & Butterfield) is made by
an Arrow of Space along
null "direction" (w-axis)
-- one-at-a-time (Phoenix Universe). Namely, a global,
atemporal, and non-Archimedean realm
of the universe (global mode of spacetime) is complemented by
a quasi-local, physical, and
teleological realm of the universe (local mode of spacetime). The
global, non-Archimedean mode keeps an intact
potential reality separated from its fleeting "jackets" (Plato),
while the local, Archimedean mode is produced as re-created "back bone" of the whole universe
at all
length scales, made by "measurements"
executed by the self-acting universe on itself -- one-at-a-time.

The
instant 'now' separates the two modes of spacetime, which evolve
along null "direction": one-instant-now-at-a-time.
Hence the topological dimensions of spacetime are being accumulated
during the Arrow of Space, and because
all "dark gaps" of the global mode of spacetime are made
zero by the "speed" of light, we observe a
perfect spacetime continuum and try to explain the dynamics
of spacetime with 'time read with a clock' due to change of
coordinates within spacetime. Hence Einstein's
total field and the 'thoughts
of the Universe' are completely missing in current GR and QM
textbooks.

Q1: "GR works
perfectly well and I can't agree with your ideas."A1: This is a statement, not question. Recall that the
current version of GR is based on "miracles": you can't have any
geometry at Planck scale. There is nothing
resembling law and order in the "spacetime
foam" to raise a robust
Lorentzian metric within 10-30 seconds "after" the "big
bang" and inflation of universe's "size" (with respect to what?) by a
factor of 1078, and keep
the Lorentzian metric for at least
13.77 billion years rooted on Planck length at which
"points" are fuzzy and locality has lost its meaning
13.77
billion years ago.

Q2: "What do you
intend
to achieve?"A2: A model of bootstrapped universe, in which every geometrical
"point" is determined by states of mater ("jackets")
that are pre-correlated with 'the rest of the universe'.
Which will be impossible if Archimedean space were fundamental
phenomenon. Hence RSP. Notice that at
Planck scale the equation SL = 1 (Small is
denoted with S, Large with L), which holds
for observer A, is again valid, but
now it describes 'the non-Archimedean universe as
ONE' of observer(s) B. Then we use this
unique ONE entity as
Reichenbach's Third Cause to explain the
relational ontology produced by entanglement
during the flow of time (cf. Escher's drawing hands
below).

Q3: "How do you
explain the
mind-body problem?"A3: With 'potential reality' in the
Arrow of Space; but please see first
Gottfried Leibniz, Wolfgang Pauli, and the
Eskimo metaphor. We
encounter two forms of potential reality: one explicated
under macroscopic conditions (BrainMonad),
and the GPI field of quantum-gravitational world (observer
B). The first interacts with the brain and the physical
world along the Arrow of Space, and can be metaphorically explained
as a 'steering wheel' of 'the car' (brain and 'the rest of the
universe'). Namely, the BrainMonad is
neither 'matter' nor 'mind', but a macroscopic form of
potential reality. One might suggest that it has a dual nature,
because it acts like a "filter" through which the Psyche can
enter spacetime, a bit like images displayed on a TV screen (not
located inside TV). So, if the BrainMonad
is a 'steering wheel', the 'driver' is the
human mind endowed with Free Will. We speculate that the GPI
field may also act as a "filter" for Universal Mind (Henry
Margenau), but cannot prove such claim. Point is, the 'driver'
(human mind) may have access to the joint 'steering wheel' (BrainMonad
& GPI field), and alter the propensities (not "probabilities") for
future potential events, almost like learning a new
motor skill. Math is a crucial issue, too.
The first off task here is to explain the physics of
binding phenomenon -- how all sensory
"data" are combined into a single experience, derived from their
joint amodal presentation (BrainMonad).
In my opinion, the only way to approach the challenge is to assume
that the whole universe as ONE works as a "brain".
Hence RS gravity.

Q4: "What is this all
about, Dimi?"A4: Tough question. Actually, it's all about cat astrology. I'm also
selling cat food from my website, with special discounts for
theoretical physicists like you. Aren't you interested?

Feel free to submit your questions. Just please don't forget
that RS theory of gravity is an
alternative to all multidimensional theories in which gravity would
operate in some "extra dimensions" with some hypothetical "gravitons".

For example, people
try to bridge the "desert between the weak and Planck scales" with
"extra compact spatial dimensions" leading to "(4 + n) dimensional
theory" in which "particles cannot freely propagate in the extra n
dimension, but must be localized to a 4
dimensional submanifold", while "the only fields propagating in the
(4 + n) dimensional bulk are the (4 + n) dimensional graviton"
(N. Arkani-Hamed, S. Dimopoulos, and G. Dvali,
arXiv:hep-ph/9803315v1, p. 2). Then comes this (ibid., p.
3):

"As within any extension of the standard model at the weak scale,
some mechanism is needed in the theory above mEW to
forbid dangerous higher dimension operators (suppressed only by mEW)
which lead to proton decay, neutral meson mixing etc. In our case,
the theory above mEW is unknown, being whatever gives a
sensible quantum theory of gravity in (4 + n) dimensions! We
therefore simply assume that these dangerous operators are not
induced."

But every sensible
quantum theory of gravity should solve the most widely known
public secret in theoretical physics --
localization. First things first.
Then comes QM and GR, and
the new Quantum Geometry in which the
geometrical points possess quasi-local structure due to the
global mode of spacetime of 'the universe as ONE',
shown with red in the drawing
below.

One can introduce "boundary" on
spacetime only and exclusively only with
RSP. This kind of "boundary" is the only possible
logical option for gravity. We are
macroscopic observers, and in the case of Archimedean geometry the
entanglement of space Espace , expressed
with "fusion" of Small and Large, is effectively zero, yet it takes
values in an open interval

Espace
Є (0, ∞).

Also, Baldy's Law, according to which “some of it plus the rest of
it is all of it,” is strictly valid only for an inanimate macroscopic world with Espace
effectively zero, but does not hold for
Quantum Gravity.
More in
A2above.

The so-called 'God's thoughts' refer to a web of
correlationsof all physical systems, which occur in a hypothetical'global mode of spacetime' produced by the
Arrow of Space.Relative to a physical (inanimate) clock, the global modeof time will inevitably look "frozen".

This "frozen" time pertains to 'the whole universe as ONE'and to the so-called BrainMonad. The "speed" of gravityis dual: both "instantaneous", in the global mode ofspacetime, and finite, in the local mode. The endresult is a quasi-local mass and quasi-instantaneouscorrelations, resembling those in a school of fish.

Relative to the local mode of spacetime, the global modeis at 'absolute rest', and serves as the reference fluid ofGeneral Relativity. It is located ]between[ any twoneighboring "points" from the spacetime manifold, andrenders its local mode a perfect continuum: due to theso-called speed of light, the "separation" of the spacetimepoints (local mode of spacetime) is in fact zero.

There is no direct link between the local and globalmodes of spacetime, i.e. between 'physical reality' and'potential reality': the UNdecidable pre-quantum KS statemust not be included in the set of its color-able explications.

The cosmic vacuum/quantum aether, called hereglobal mode of spacetime, is an absolute referenceframe pertaining to the whole universe as ONE: the physical world there is in absolute rest, in thesense that its proper time is zero, as "seen" fromsuch luxonic reference frame.

The current GW detectors are manifestly
blind and deaf tothe ripples of spacetime. Their proper detectors mustbe endowed with the faculty of 'self-acting', resemblingthe
human brain. Ditto for the so-called "dark energy".

"In the first place, we entirely shun the vague word "space," of which, we must honestly acknowledge, we cannot form the slightest conception."

Albert Einstein, Relativity: The Special and General Theory, 1920"According to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an ether. But this ether may not be thought of as endowed with the quality characteristic of ponderable media, as consisting of parts which may be tracked through time."

A. Einstein, Äther und Relativitätstheorie, May 5, 1920

(Lisa M. Dolling et al., The Tests of Time: Readings in the Development of Physical Theory, Princeton University Press, Princeton, 2003, p. 346)

It is suggested that the spacetime manifold is a dynamical entity re-created at every step of a hypothetical spacetime arrow; the latter is due to the "expansion" of space by the dynamic dark energy (DDE). Two modes of spacetime are postulated in this Phoenix Universe: local mode, in which we have point-like events cast on a perfect continuum, with a "carpe diem" unit probability (cf. the measurement problem in QM here), and a global atemporal mode, in which a Machian-type negotiation of every next step is being processed. The effects from the global mode of spacetime, which literally build up '3-D space', begin from the macro-scale of classical physics in two "directions": quantum effects toward the Small, and dark matter & dark energy effects toward the Large. Briefly, we model the universe as a huge brain which 'thinks' with its global-mode state by following the rule 'think globally, act locally'. The implications for quantum gravity are explained by revealing the two modes of spacetime in Quantum Theory and General Relativity, and by suggesting conceptual solutions to the problems and paradoxes hindering the quantum gravity of He Who Does Not Play Dice.

We haven't the money, so we've got to think!Lord Rutherford, 1962 Brunel Lecture, 14 February 1962

Overfunded research is like heroin: It makes one addicted, weakens the mind and furthers prostitution.Johann A. Makowsky, The Jerusalem Post 19.4.85

Regarding Quantum Mechanics (QM), the aim is to avoid the incomprehensible paradoxes and artifacts in it (watch the double slit experiment here), which originate from its textbook interpretation (Niels Borh's belief that quantum world can only be "seen" through classical "glasses"). We can indeed understand the quantum world (but not the current QM textbooks; cf. Richard Feynman), by changing the "glasses" through which we "see" and construe the quantum world: the universe modeled as a brain.

Regarding the artifacts in QM, we may be in a situation similar to an Eskimo trying very hard to comprehend the notion of "trunk". In our case, we encounter an incomprehensible wave-particle complementarity, which could be just an artifact from our wrong thinking, like the "nose-arm complementarity" in the case of the Eskimo observing elephant's trunk. Surely 'the quantum system' doesn't live in any relativistic space, so one should expect all sorts of headaches and artifacts (e.g., "diese verdammte Quantenspringerei", Erwin Schrödinger) from imposing wrong "glasses" onto the quantum world 'out there'. (A typical example is Franco Selleri's quest for detecting de Broglie waves.)

The prerequisites for the interpretation of QM suggested here originate from Schrödinger. Back in 1935, Erwin Schrödinger stressed the following:

"The rejection of realism has logical consequences. In general, a variable has no definite value before I measure it; then measuring it does not mean ascertaining the value that it has."

And in a letter to Einstein dated 18 November 1950 (quoted after J. Bub, p. 115), he wrote (emphasis added):

“It seems to me that the concept of probability is terribly mishandled these days. Probability surely has as its substance a statement as to whether something is or is not the case — an uncertain statement, to be sure. But nevertheless it has meaning only if one is indeed convinced that the something in question quite definitely is or is not the case. A probabilistic assertion presupposes the full reality of its subject.”

We are obviously dealing with a new form of reality: a probabilistic assertion, and the Hilbert space itself, cannot accommodate the Kochen-Specker case in which "the something in question" is beyond an unequivocal "is or is not" state (cf. "an unequivocal true-false value" in Isham and Butterfield, p. 3; emphasis added):

Check out the implications of KS Theorem to the Precise Value Principle (PVP) and the statistical interpretation of QM from R.I.G. Hughes, p. 164.

Going back to Ernst Specker's tripod, if there are states of the tripod in which one of its legs has UNdecidable color, then not just this particular leg, but the whole tripod will be UNdecidable. It's a package.

The phrase "an incomplete Kochen-Specker colouring" (Helena Granström, p. 2) has no meaning whatsoever; it is the result from imposing wrong "glasses" onto the quantum world. And if you subscribe to the modern quantum mysticism -- "the quantum state is not a physical object, it is a representation of our state of knowledge, or belief" (Itamar Pitowsky, p. 28) -- your brain will wind up in a schizophrenic state of, say, 68% "knowledge" of the quantum state, and 32% of "[what da heck is that uncolored KS sphere?]".

No mental concepts, such as 'knowledge' or 'imagination', are admissible in the ontology of quantum reality. We must never mix apples with oranges (Res Extensa and Res Cogitans).

Karl Svozil refers to this UNdecidable faculty of the quantum world as "ambiguity" (p. 4), and stressed: "This ambiguity gets worse as the number of particles increases." If you think about the quantum world with classical concepts, it will get from bad to worse, until you end up with the (old) cosmological "constant" problem (more on that from Alan Guth).

As Erwin Schrödinger might have said in 1935, the same "variable" that has had no definite value before you measured it will continue to keep its UNdecidable nature after you "measure" it as well. And you can't fit it in any Hilbert space, of course (what is the dim(H) for 32% "uncolored" and 68% colored KS sphere?).

Let's give it a name: potential reality. In the quantum realm, it (i) offers its context-dependent explications (a.k.a. "observables"), (ii) keeps the sameness (Genidentität, Kurt Lewin) of particles of the same type, which MTW regard as "a central mystery of physics" (p. 1215), and (iii) facilitates the ultimate quantum phenomenon: entanglement. It may be difficult to grasp, but is much simpler to the juggling with the possible implications from Bell's inequality and their loopholes (e.g., Ghim and Zhang).

As to quantum gravity (notice the opinion of an expert here), the 'potential reality' is introduced to revive the physical objectivity of spacetime "points", by making the spacetime manifold itself an emergent phenomenon: "The requirement of general covariance takes away from space and time the last remnant of physical objectivity" (A. Einstein, Grundlage der allgemeinen Relativitätstheorie, Annalen der Physik 49 (1916) 769-822). The current formulation of GR can only determine "the mutual relations that exist between the gravitational field and the matter fields (i.e. the value the gravitational field takes where the matter field takes such and such value)", but not "the proper time between spacetime points" (Wiki; more from Butterfield and Isham), and the field equations "cannot even uniquely determine the topology of a manifold" (Alan Macdonald, Einstein's Hole Argument, p. 4).

How can we fix these problems? By introducing two connections, geometric (local mode) and torsion (global mode of spacetime). The torsion connection is completely vanished (Hehl and Obukhov) in the local mode, hence its effect are considered "dark" (see Alex Murphy).Notice also that a hierarchy of 'potential reality' (never in plural), resembling the structure of cognitive concepts, is postulated (application here). In metaphysical terms, it supports the views of Aristotle and Spinoza: no "parts" of the infinite can exist, as the infinite Substance is indivisible. It's not like Russian dolls.

In the case of a human brain (not mind or consciousness), the UNspeakable potential reality can be explicated with, for example, three (and many more) sayings, which produce "measurements" (if you prefer QM jargon) on it:

None of these "measurements" can "collapse" the potential reality (Platonic idea) explicated with these sayings. And if our brain can work with 'potential reality', so can the universe modeled as a brain.

Regarding the brain above your neck: its 'potential reality' is to its human self what EM radiation is to your subjective experience of 'color' -- no direct action of the human self on its brain is allowed in science. Hence it may be conceivable that the potential reality entwined with the human brain could be bridged to that of the quantum vacuum (BCCP). All you may need is an arrow of spacetime. (Notice, however, the 'two rules of success' here.)Notice also that 'potential reality' is an intrinsically holistic phenomenon, so it will be very misleading to call it "dark", just because it is not possible to be traced back from any of its physical explications in the r.h.s. of Einstein field equation.

The sole "explanation" of this ultimate puzzle of GR, offered by Chris Isham, was that, "after all, general relativity does seem to work well as a theory, and yet I can certainly read the time on my wrist watch!" But your wristwatch should not be able to read more than one instant from this (global) non-linear "time". Complex problems have simple, easy-to-understand wrong answers (Murphy's Law No. 15).

We need quantum gravity, to uncover "the proper time [tau] along spacetime trajectories" (Carlo Rovelli; drawings here) and the genuine 'time variable' associated with the expansion of space due to DDE. Once we achieve this formidable task, we will (hopefully) find out what -- if anything -- might remain unchanged/conserved in this particular (global mode of) time. In the current 'GR without DDE', the dynamics is "entirely generated by constraints. The dynamical data do not explicitly include a time variable" (Karel Kuchar), and one can only hope that "the energy momentum tensor which is the source of gravity" might be conserved "due to the Bianchi identities" (E. Guendelman, p. 9). Such hopes are (i) utterly murky even in the textbook GR, because any observable of the gravitational field is "necessarily quasi-local" (Laszlo Szabados) and (ii) not applicable to the new dynamics of GR due to DDE. For example, Noether's Theorem holds only for flat spacetime, and the Bianchi identities are applicable only for 'GR without DDE'.

"... the Einstein equations of general relativity are local equations, relating some aspects (notice the poetry -- D.C.) of the spacetime curvature at a point to the presence of stress-energy at that point. Additionally, one also has local chronology protection, inherited from the fact (Sic! -- D.C.) that the spacetime is locally Minkowski (the Einstein Equivalence Principle), and so “in the small” (that's the sole application of those 'twice-contracted Bianchi identities', cf. George F R Ellis and Henk van Elst, Eq. 2 -- D.C.) general relativity respects all of the causality constraints of special relativity.

"What general relativity does not do is to provide any natural way of imposing global constraints on the spacetime — certainly the Einstein equations provide no such nonlocal constraint."

Which is precisely the missing element needed to address the 'global properties of spacetime' in the presence of DDE. As R. Rakhi and K. Indulekha acknowledged (p. 5): "Because this energy is a property of space itself, it would not be diluted as space expands. As more space comes into existence (the same phrase was used by Sean Carroll -- D.C.), more of this energy-of-space would appear. (...) So the mystery continues."

The intrinsic dynamics of the phenomenon of 'more space comes into existence' could only be detected in a reference frame from which this phenomenon is excluded, like 'not acting there'. But because DDE acts on the whole 3-D space en bloc, without any irregularities, there is no place in the universe in which we could install a clock and say -- look, this clock does read the (global mode of) time pertaining to the omnipresent phenomenon of 'more space comes into existence'. It is the theory of relativity itself, which cannot "detect" it. Which is why some people call it "dark".

Although the mathematical meaning of the phrase "more space comes into existence" is not clear, I don't think Kurt Gödel would have called the ultimate source of spacetime points "dark". Once we move from physics to geometry, it's a whole new world there. Pity Mike Turner called it "dark"; that's so wrong!

Notice also that Matt Visser (see above) considers the metaphysical assumption that the spacetime were "locally Minkowski (the Einstein Equivalence Principle)" to represent a fact. But we only have a mathematical fact that locally , "over" a point, one can indeed eliminate the gravitational "field" by hand (Hermann Weyl). But because nobody has so far explained the resulting quasi-local nature of the gravitational field's observables (see Laszlo Szabados above), nor the origin and the mechanism of inertial reaction "forces", I think we should be very cautious and open-minded in interpreting such mathematical facts and theorems, like the above-mentioned Bianchi identity.

If you are looking for a genuine quantum-gravitational measuring device, your wristwatch (as well as the one of Kip Thorne) fits the bill, because it reads an already-linearized (see the explanation of 'already' below) time variable obtained from the "the proper time [tau] along spacetime trajectories" (Carlo Rovelli) and the (global mode of) time associated with the expansion of space due to DDE.

This is to me the ultimate puzzle in present-day GR. The sole "explanation" of Chris Isham was that, "after all, general relativity does seem to work well as a theory, and yet I can certainly read the time on my wrist watch!"

I will desist to comment on C. Isham's observation, and will instead take the liberty of being (again) deadly boring, by explaining the difference between 'GR without DDE' vs 'GR with DDE'.

First, a simple example from STR, with a trajectory of a Frisbee, on the fixed background of Minkowski spacetime: we can calculate the instantaneous state of the Frisbee at each point from the trajectory, and attach to this dimensionless point a well-defined vector. My teenage daughter couldn't understanding how it is possible to attach a vector to a "point", and I explained the puzzle by saying that the information from a finite interval from the history of this infinitesimal "point" is encapsulated in it -- we instruct this interval to shrink asymptotically toward zero -- so the vector is indeed well-defined. All this is possible under the premise of the fixed "grid" in STR. But once we move to 'GR without DDE', the "grid" is gone (Emilio Elizalde): at each and every "point" from the trajectory, the non-linear mutual determination of matter and space (John Wheeler) takes place. But in what time (see Chris Isham's wrist watch above)? There is no background grid or "ether" w.r.t.w. one can define the dynamics of 'GR without DDE'. Yet people don't ask such questions and prefer to just do calculations with the linearized approximation of 'GR without DDE'. As another expert explained to his undergraduates, "one begins by introducing the notion of a tangent vector to describe an infinitesimal displacement about a point p " (Bob Wald, p. 4). But again, in order to recover the true dynamics of 'GR without DDE', you need some "ether" or rather 'reference fluid' w.r.t.w. one can describe the fundamental phenomenon of transience (Abner Shimony): see the so-called Aristotelian Connection here.

At this point, the 'GR with DDE' comes to rescue theHamiltonian formulationof 'GR without DDE': we have a brand new, global degree of freedom of spacetime en bloc, hence can recover the transience of spacetime, as driven by the source of DDE, along the arrow of spacetime -- see above.

To identify this same source in the quantum realm (called 'potential reality'), let's go back to the interpretation of QM here.

In a nutshell, the so-called PR2 interpretation of QM offers a solution to the non-unitary "collapse" by replacing the alleged "U" and "R" processes with a new (at least to people like Ed Witten, Steven Weinberg and Gerardus 't Hooft) form of reality, known since Plato, called here 'potential reality'. Its quantum presentation is ubiquitous and has zero entropy; hence the familiar notion of time, which pertains to monotonic increase of entropy "in time", is not applicable to 'potential reality'. It may produce "shadows" in terms of 'quantum observables', yet these "shadows" cannot be traced back, to reveal any evidence of such events (or "quantum information") ever having emerged from 'potential reality'. It may act, yet not experience any backaction from its "shadows". Hence it is the ultimate "background" for QM and GR. It evolves along the arrow of spacetime (resembling the "memory" of the universe), by unfolding from 'the ideal monad without windows'. In the local mode of spacetime, every "point" is filled with an already explicated value of its 'potential reality', and as the latter evolves and becomes enriched, the "number" of its localized explications increases accordingly: more and more things come into existence in the universe. As John Wheeler put it, "Time is Nature's way to keep everything from happening all at once". Only with 'potential reality' there is no need nor place for any 'unitary dynamics' in the local mode of spacetime. This is the metaphysics of 'the universe modeled as a brain', viz. the interpretation of 'expansion of space' along the arrow of spacetime.

From this perspective, if we follow the deflation time arrow in the local mode of spacetime, things will gradually fade away in a strictly non-unitary fashion, but will only approach asymptotically The Beginning, because it is logically impossible to reach It from/within the local mode of spacetime (cf. the paradox here). The solution is 'dual age cosmology'.As M. Gell-Mann and J. B. Hartle have noticed, "quantum mechanics is best and most fundamentally understood in the framework of quantum cosmology" (quoted after Claus Kiefer, p. 1). And vice versa.

The current situation with inflationary cosmology was presented by Alan H. Guth, the winner of the 2009 Isaac Newton medal, on 13 October 2009. He speculated that our galaxy could be an amplified "quantum fluctuation" (Part 2, 3:30 - 6:16),

... and then confessed his "Nightmare of Dark Energy" (Part 2, 11:36), on which the "eureka" of inflationary cosmology is grounded.

Perhaps Alan Guth should have entitled his talk 'The Dark Energy Nightmares of Inflationary Cosmology', as encapsulated in the [?] area in V. Mukhanov's paper:

With the so-called dual age cosmology, the "first" Plankian time "after" The Beginning is stretched toward infinity in the local mode of time. This is otherwise being explained as 'the universe started asymptotically from time zero'. In this local mode of time (read by your wristwatch), the universe is effectively eternal and its age is indecisive, while in the global mode of time it has a finite age. Another nice feature ofdual age cosmology is that The Beginning is an atemporal phenomenon, and is always present (global mode of time) in the instant 'now'.

Let's go back to QM. Think of a trajectory of a Frisbee: at each "point" from the trajectory (the latter could be a perfect continuum of such "points"), the state of the Frisbee is a perfectly well defined fact, with a 'carpe diem' unit probability. I use this simple idea to introduce a new kind of cancellation mechanism, pertinent to 'potential reality', for removing the non-unitary "R" process (historical account from October 2002 here); hence the motto of this web site, since July 1997: Dead matter makes quantum jumps; the living-and-quantum matter is smarter.

Notice that not only the "R" process is being removed (as did Hugh Everett III in his relative state interpretation of QM), but the alleged "U" process and Hilbert space as well.

The established theoretical physics community, particularly Niels Bohr, ridiculed Everett's interpretation to the extent to which Hugh Everett left physics (and became millionaire).

The next step is to elaborate on the continuum of such explicated quantum states (local mode of spacetime), which emerges from the back bone of the whole physical world (global mode). Here we enter quantum gravity, particularly the emergence of time and space from "something else" (Isham and Butterfield): check out the arrow of spacetime here, and think of 'the whole universe' as a huge brain which self-determines its consecutive quantum-gravitational states in line with the Bootstrap Principle of Geoffrey Chew. Also, the alleged quantum "fluctuations" are interpreted as flexibility of 'the quantum state' to offer potential states, one of which to be chosen (one by one) by 'everything else in the universe'. Hence the speculation about some "spacetime foam" and the insoluble problems from it (how do you tell apart spacelike from timelike at Planck scale?) are avoided from the outset. Possible practical implications are outlined here.As in the case of Hugh Everett, many theoretical physicists jumped to play the role of Niels Bohr, chief among them was Britain's leading expert in quantum gravity, who declared seven years ago (Wed, 23 Oct 2002 19:24:15 +0100):

"You do not know enough theoretical physics to help with any research in that area."

To the best of my knowledge, nobody has so far offered some new cancellation mechanism as a joint solution to the measurement problem of QM and the cosmological "constant" problems. Such cancellation mechanism is built in the UNdecidable quantum state (quantum presentation of Platonic ideas) from the outset: instead of dealing with some |alive cat> & |dead cat> from the "U" process, we encounter an UNdecidable 'cat per se', so if we happen to observe an 'alive cat' as a fact, with unit probability, the 'dead cat' will happily live undisturbed in 'the UNdecidable quantum state'. No "collapse" nor dead cat doppelgänger à la Everett are needed.

The animated cat above is very deceptive, because its "evolution" prior to the "collapse" seems fully deterministic. If you think about the quantum "evolution" of such superposed states in terms of 'energy eigenstates', notice the imaginary unit in S. Carroll's essay, Eq. 4: "all of the time evolution is encoded in the phases [XXX]" (ibid., p. 6). But how do you encode 'time evolution' in complex phases (Chen Ning Yang)? Recall that, after you "collapse" the cat, the alleged "time parameter" in the Schrödinger equation turns into some non-relativistic [you-name-it], but the equation itself doesn't say anything whatsoever about the nature of this "time parameter" prior to the "collapse", as introduced by hand (along with the Born rule) in QM textbooks.

To understand the origin of this whole mess, read the second sentence from the excerpt below (R.I.G. Hughes, The structure and interpretation of quantum mechanics, p. 77):

This "general principle", however, does not cover the case of UNdecidable, hence not-yet-physical, KS quantum state, as explained by R.I.G. Hughes here. One can hardly overestimate the enormous confusion from this huge blank spot in current QM textbooks -- just recall the ongoing quest for "quantum computing" and the alleged "decoherent histories" (J. Halliwell; cf. Franck Laloë, Sec. 6.4 and footnote 47 on p. 81).

All these problems are resolved in the PR2 interpretation of QM from the outset. Moreover, the new cancellation mechanismis introduced to explain the ongoing, as-we-speak mechanism of 'the flatness problem' (asymptotically flat spacetime conjecture), producing an extremely precise balance between the two tug-of-war effects, CDM & DDE, of the geometry of spacetime at cosmological scales.

Otherwise we have to inject up to 96% "dark stuff" (with all sorts of "ghosts") into the current theoretical physics, as calculated under the assumption that 'potential reality' doesn't exist. As Evalyn Gates put it (p. 196), the detection of DDE was "like finding an elephant on top of a table impeccably set with the finest china and silver (...). We stare in shock at the uninvited guest and demand to know where the elephant came from -- and how it got into (the) room."

Regarding the arrow of spacetime: notice that the Frisbee-like sequence of explicated facts (local mode of spacetime) is inherently background-free, because the background (the reference fluid of GR and the UNdecidable quantum state) is 'not there', being placed ]between[ the "points" of the continuum of the local mode of spacetime. In other words, the "dark gaps" of the global mode are not like the real gaps between the tiles in R. Penrose's bathroom: regardless of how small tiles you choose, if you decrease the size of the tiles in a Fibonacci sequence, you will never ever reach the "gaps" from/within the local mode of spacetime (more on Fibonacci here).

This proposal makes the local mode of spacetime a perfect 3-D continuum with dynamical topology of 'asymptotically flat spacetime'. The "dark" gaps of the postulated global mode of spacetime are completely sealed off by the arrow of spacetime -- the mechanism by which the "dark" gaps of the global mode are made nonexistent in the local mode is the same that makes the "speed" of light a fundamental constant (and also hides the mirror tachyonic world). This renders the Schlaefli conjecture (L. Schlaefli, Ann. di Mat. 5 (1873) 170), as well as all "branes" and other multidimensional superstitions in GR and string hypotheses (Lisa Randall; see also A. Vilenkin below) redundant, to say the least.

In the context of GR, the global mode of spacetime is located "within" each and every point from spacelike hypersurfaces, as it "lives" exclusively on null hypersurfaces. (Unlike the geometry of spacelike hypersurfaces, the geometry of null hypersurfaces is not metric (D. C. Robinson), which leaves a challenging opportunity to introduce an additional, to the Christoffel connection,global torsion connection.) To be precise, the "duration" of the global mode, recorded with a physical clock (local mode), matches the "duration" of the atemporal "handshaking" transaction in Cramer's interpretation of QM: it has been already completed at each and every instant we "look" at it (see below).

Notice also that the hypothetical global mode of spacetime cannot be read by a physical clock (it will "stand still"). It is introduced to replace the "external time parameter" in H.-D. Zeh (p. 13) and the “auxiliary internal time” (cf. Macias and Quevedo, p. 8) by 'the reference fluid of GR'. The latter can "act" upon matter without being affected in turn by matter. In this unphysical "absolute" reference frame, an electromagnetic radiation field can indeed "stand still" (recall that EM radiation field cannot stand still with respect to any physical observer, Bahram Mashhoon, p. 14).

The next metaphysical idea is straightforward: 'time' does not originate from 'change in space' but from 'change of space'. Only if you have the latter (global mode), you may introduce the former, as 'time read by a clock' (local mode). Why? Because one cannot insert the "dark" energy of Xinto its consequence: accelerated expansion of space. The dynamics of 'the change of space' is defined relative to the "omnipresent ether X " (global mode of spacetime). The latter is located "within" each and every point from the local mode of spacetime, and is wrapping the local mode by two (in fact, one) 'numerically finite but physically unattainable Aristotelian boundaries'.

Recall what William Kingdon Clifford claimed in his paper ‘On the Space-Theory of Matter’, presented to the Cambridge Philosophical Society on February 21, 1870 (quoted after Domenico Giulini, p. 2):

I believe Clifford's idea in (2), about "the manner of a wave", is amended here with the proposed origin of quantum-and-gravitational "waves": the continuous passage of "curved or distorted" from one point (not "portion", as in (1) above) to the next one is what the arrow of spacetime does on the perfect continuum of the local mode of spacetime (the "dark" gaps of the passage are being completely sealed off by the arrow of spacetime).But why 'arrow of spacetime'? Because our good old 3-D space is not like a huge static warehouse, in which we would notice some redshifted light from moving objects, receding from us in line with the Hubble Law (some balloon metaphors may be highly misleading). It is the other way around: distant galaxies are not "speeding up" with respect to us in some absolute static space, but the very metric of space is "expanding" (I firmly disapprove of this notion of "expansion", and have suggested the so-called 'relative
scale principle').

Hence the space itself is endowed with dynamics, but then we need some ether w.r.t.w. such 'global dynamics of 3-D space' can be formulated. Then the only possibility -- trust me, there is no other option -- is to place the omnipresent ether, as 'the source of the "dark" energy', in the global mode of spacetime. Otherwise you will have to define the dynamics of space w.r.t. itself, and will look like Baron Munchausen. That's why we need an arrow of spacetime, in my opinion (but notice the opinion of Chris Isham above).

NB: If this is the case chosen by Nature, then any approach to quantum gravity, based on the "splitting" of spacetime (Brett Bolen), will necessarily be wrong.The speculations of Roger Penrose will necessarily be wrong as well: "The fuzzy idea of where and what is infinity was clarified and made more specific by the work of Penrose [45, 46] with the introduction of the conformal compactification (via the rescaling of the metric) of spacetime, whereby infinity was added as a boundary and brought into a finite spacetime region." (Ted Newman et al.)

Perhaps we may have to develop new mathematical theory of 'potential reality', such that the "state space" of Margenau's Onta (quantum presentation of Platonic ideas) would match the structure of cognitive concepts; notice that in the "cheating on 20 questions" the answer 'cloud' was explicated by a Baeysian learning rule (not the Born rule). For comparison, the categorification of Feynman diagrams requires "black boxes with many wires going in and many wires going out" (Baez and Lauda, p. 16), while in our case all wires are "instantaneously" (global mode of spacetime; see the Escher drawing below) keeping track of all virtual 'black boxes' as well (relational ontology), in order to dynamically adjust to the changing context of the game, until they jointly select the final, explicated 'black box': 'cloud' (see also the four dice here).

NB: In the local mode of spacetime, the "duration" of the total negotiation with 'everything else in the universe' (relational ontology) is zero. Hence a wave pattern is being created, without any source of these "waves" being present in the local mode, and a new form of retarded causality (biocausality; see below) can be postulated -- a revitalization of Leibnitz' harmonia praestabilita, Jung's Synchronicity, and Einstein's Überkausalität.

We definitely need mathematical theory of 'potential reality'. The task is highly non-trivial, but once we unravel the correct mathematical theory, the astonishing effectiveness of mathematics (Eugene Wigner) may drive us closer to the true quantum gravity of He Who Does Not Play Dice -- the world is not deterministic but flexible, and the 'chooser' of one possibility (one at a time) amongst infinitely many is 'the whole universe' in its state of ONE.

Perhaps the situation will improve in 2010, after the sixth consecutive failure of LIGO Scientific Collaboration (LSC) to detect GWs with the so-called "enhanced LIGO". GW energy transfer is fundamentally non-linear phenomenon (Hermann Bondi), but is wiped out with the "linearized approximation" adopted by LSC. Also, GW energy is intrinsically quasi-local, in the sense that GWs do not propagate exclusively "in one direction only" (“when the waves are all moving in the same direction”, cf. P.A.M. Dirac, Ch. 33, p. 64), as they also have a holistic global component (atemporal "handshake"), which covers the whole 3-D space en bloc.LIGO Scientific Collaboration (LSC) stubbornly refuse to acknowledge that there are no bans whatsoever on the dipole radiation, simply because
conservation of gravitational mass-energy and momentum, in a world dominated by an evolving cosmological "constant", is a wishful thinking. They fear to even mention the implications of DDE to their project, and have never tried to address the (old) cosmological "constant" problem, which inevitably occurs if you trust the unwarranted assumption that the "dark energy" from the quantum vacuum can only spring from Lorentz-invariant stuff with positive energy density.

Is it possible to detect some 'elementary shift' of the expanding metric -- the "intrinsic time interval associated to any timelike displacement", T. Jacobson, pp. 18-19 -- due to the omnipresent and perfectly smooth DDE? In what reference frame? Notice that you're dealing with some "fluid" that "has zero inertial mass! It can be accelerated with no cost, no effort" (B. Schutz, p. 255) and "provides an all-pervading energy density and negative pressure that are the same to all observers, at all places, and at all times in the history of any universe model, even the expanding ones." (p. 257)

The insane efforts of LSC remind me of the old joke about a drunken man, who has lost his key somewhere in the dark, but is searching for it under the street lamp, simply because it is brighter there. Only LSC's "key" costs billions. Which is why I accused LSC members of aggressive professional negligence, and offered them to review my White Paper. They responded with dark silence, as usual.

"Gravity is a different beast from radiation of any kind. Being mediated by virtual particles, which may be considered to be kept permanently virtual by the physical non-existence of gravity shields or absorbers, gravity can act (nonlocally) with infinite speed -- in effect, with precognition. That is exactly what it does, if Mach's principle has any substance. The fixed stars "know" the subway is going to jerk, because they have sent their virtual spies forward in time to find out about it."The crux of the idea of biocausality is well-known. Back in 1953, Wolfgang Pauli suggested that the concept of finality ("the end (telos), that for which a thing is done", Aristotle, Physics 194b33) should be considered as a complement to causality in deterministic and statistical laws. The stipulation here is that Aristotle's effective cause & final cause determinate jointly the next state of all material constituents of the universe relationally, in line with the bootstrap principle 'think globally, act locally' (global mode of spacetime; see the Escher drawing below), but in the local mode the resulting biocausality is retarded, along the arrow of spacetime.

Some history of the project outlined above. In January 1972, shortly after my demobilization from BG army (age 19, with the lowest possible rank), I decided to study psychology and "sort out" the physics of the human brain, to explain and eventually explore its amazing latent abilities. It took me sixteen years to realize that the task is unfeasible with the established Weltbild, and on 22 May 1988, at 23:45 local time, I decided to try the project 'the other way around', namely, to develop a model of the universe as a 'brain', such that there will be a natural explanation of the physics of the (small) brain from the outset. After many trials and errors, the first (relatively) encouraging evidence appeared in July 1998, but the project is still far from being completed. I am trying to find any weak points or inconsistencies in it, just as I would do with a brand new parachute, which I would have to put on ultimate test by jumping from a helicopter. I do like the unfolding of the whole project, since it seems to me that all pieces of 'the bridge' snap to their places effortlessly. Yet there is no room for contention, and I never go into the mood of 'chi si contenta gode' (a contented mind is a perpetual feast; he who contents himself, enjoys).

Hopefully, if we join our efforts and knowledge (included at the level of our collective unconsciousness, cf. Jungian Kollektives Unbewusstes), some day we may succeed.

Meanwhile, please keep in mind the prediction of Robert Millikan, Nobel Prize in Physics (1923): "There is no likelihood man can ever tap the power of the atom."

This web site is my feedback to all people, who are helping me, one way or another, with improving the theory. Feel free to download it (app. 11.8MB) from

By unzipping PHI_info.zip, a new folder, !Einstein_PHI , will be created on your hard drive. Find there START.html.lnk and open it (it is linked to this front page (index.html) in the website folder).

If, for some reason, you believe we don't need new approach to quantum gravity, try to understand Sean Carroll's speculations (448 pages, January 2010):

"... wavefunctions appear to collapse in one direction of time but not the other is not an explanation for the arrow of time, but in fact a consequence of it. The low-entropy early universe was in something close to a pure state, which enabled countless "branchings" as it evolved into the future." Yet he acknowledged: "we can't, once again, define a conserved total energy in any reasonable way." Consequently, the main speculation of S. Carroll about some "low-entropy early universe" is just as unclear as is its gravitational energy which would evolve "into the future". (Also, there may be a critical low geometric entropy state of the 'extremely early' universe, which may blur the timelike and spacelike directions themselves, and turn them into some primordial quantum dough from which nothing could possibly "decohere", ever.) As one of his senior colleagues summarized, "the magnitude of the entropy of the universe as a function of time is a very interesting problem for cosmology, but to suggest that a law of physics depends on it is sheer nonsense."

"... when treated in a multiverse framework, fundamental time is directionless and consequently physical laws inherit its time-reversal symmetry. Despite that reversal symmetry is broken for the local time by the bubble nucleation, the bubble still inherits laws of physics at birth from the multiverse, without modification. Thus the emergent time’s arrow in the bubble does not affect the time-reversal symmetry imprinted onto the physical laws that the bubble inherits from birth in the multiverse. (...) An emerging time in the multiverse does not appear plausible since the emergence adds information on the multiverse that wasn’t there prior (... but we face the same kind of emergent non-unitary phenomenon with DDE -- an evolving cosmological "constant" that springs from the quantum vacuum - D.C.)."

"During the last 25 years a new scientific paradigm gradually emerged. (...) My main goal is to learn how to make scientific predictions in this complicated framework."

Sure enough, Andrei Linde tried to answer the first off question of how many universes are in the "multiverse" (arXiv:0910.1589v2):

Coincidently or not, his calculation matches the number of angels (mostly blond) that can fit on the head of a pin, as suggested by A. Linde's medieval colleagues.If these people were here in the 21st century, I suppose they would gladly join LIGO Scientific Collaboration (see above), and speculate that, shortly after the "big bang", the whole universe was still small enough to fit on the head of a pin, with strong ripples of the spacetime metric (see their picture above). They would love to feel again like an 'absolute observer' with an absolute clock paired with an absolute measuring rod, to tell the dimensions of the universe and its current cosmological age, as read by their absolute wristwatch. But instead of arguing about angels (either blond or not), they would certainly prefer to measure the "ripples of the metric" (albeit very weak), which again can only be seen from the same standpoint of 'absolute observer'. Needless to say, they will easily publish tons of articles on "GW astronomy", and will ignore any alternative viewpoint on what can be "seen" by such absolute observer: nothing but "gauge-dependent" stuff.

Back in April 1986, Yakov Zel'dovich wrote in a letter the following (private communication): "Long time ago, there was a period of time during which there was still no time at all." Of course he was joking.

Yet the 'global mode of time', pertaining to potential reality, cannot be read by any physical clock, because the poor inanimate clock will read it as 'stand still' or "no time at all", as Yakov Zel'dovich put it. Maybe The Beginning, which lives in "no time at all", is always with us (dual age cosmology).

Anyway. Perhaps in the next twenty-five years a new paradigm will emerge: the universe modeled as a brain. The driving force of its arrow of spacetime cannot spring entirely and exclusively from the "brain" itself, which brings us to Virgil's statement: Mens agitat molem (The Aeneid, Ch. 6, 727).

In German, it reads: Der Geist bewegt die Materie. Physically, Der Geist may look like 'the ideal monad without windows'. However, no scientific predictions can be made about it, or else we will conflate religion with science. Thank God, this is impossible.

"Leibniz thought the idea of God as an astronomical maintenance man as absurd. He believed that God had carefully chosen among an infinity of possible worlds, the one He felt the most suitable. So that although we may not have a perfect world, it was the

"best of all possible worlds."

I intend, Deo volente, to talk on a similar subject on 25 November 2015,

Suppose someone claims that *the real cows are indeed round*, because cows might be approximated as spherical objects, and then ask your government to allocate a significant portion from your taxes for detecting the unique pattern of 'real spherical cows'.

Likewise, you were told by a number of people that, after applying their spherical-cow approximation to Einstein's theory of general relativity, they might eventually detect gravitational waves (GWs): "we haven't been able to detect them yet because they are very weak" (Jim Hough).

However, their persistent optimism is rooted on artifacts due to their spherical-cow (=linearized) approximation of GR,

I fact, they ignore all problems due to their approximation. For example, Hermann Weyl proved in 1944 that such spherical-cow approximation implies the existence of a tensor that, except for the trivial case of being precisely zero, does not otherwise exist,

Regarding Ralph Cordey at Astrium UK and the so-called LISA Pathfinder: How much this spherical cow will cost to UK taxpayers, I wonder.

Yours sincerely,

Dimi Chakalov35 Sutherland StSW1V 4JU----------

Note:The failures to detect GWs were "explained" by Jim Hough with a very misleading statement: "we haven't been able to detect them yet because they are very weak". In fact, GWs are immensely powerful phenomena, but nobody -- Jim Hough and Sheila Rowan included -- can offer a non-linear theory of GWs. All they can do is to imagine that, by the time GWs reach LIGO or LISA, they will be "very weak", such that their spherical-cow approximation to Einstein's GR would be correct. But again, they don't have any non-linear theory of strong GWs, fromwhich some "weak limit" can be derived.

All they do is asking for more taxpayers' money for detecting spherical cows, instead of doing their homework first on paper, to demonstrate such "weak limit" to initially strong GWs. Pity nobody cares.

"The first question we have to answer is: What are the base states for the system? Now the question has been put incorrectly. There is no such thing as “the” base states, because, of course, the set of base states you may choose is not unique. New sets can always be made out of linear combinations of the old. There are always many choices for the base states, and among them, any choice is equally legitimate. So the question is not what is the base set, but what could a base set be? We can choose any one we wish for our own convenience. It is usually best to start with a base set which is physically the clearest. It may not be the solution to any problem, or may not have any direct importance, but it will generally make it easier to understand what is going on."

There is such thing as “the” base state (never in plural) -- the UNdecidable KS state, as explained below.

... are just possible physical manifestations of “the” base state, much like the three sayings above, emanating from their UNspeakable potential-reality state of your brain.

Thus, the system {“the” base state & |whatever>} evolves along the arrow of spacetime in a strictly quasi-local fashion: see Fig. 1 and Fig. 2 below.

Feynman also stressed [Ref. 1, 12-2]: "That’s the question: How do the amplitudes change with time in a particular (fixed) base?"

The amplitudes change along the global mode of time: see again Fig. 2 below. Don't be befuddled by the anti-relativistic "time parameter" in the Schrödinger equation, because you need “the” base state to eliminate the measurement "problem" in QM and reconcile QM with STR, as well as solve the puzzle of quantum vacuum and gravitation: “the” base statedoes not gravitate.

It would be nice if Robert Geroch writes up a sequel to his 30-year old book General Relativity from A to B, entitled "General Relativity from A to A+ds", to elucidate the concept of 'interval' in GR -- professionally. But he wouldn't. Trying to discover new math is a tough challenge.

The last time I heard from Robert Geroch was eight years ago, only to require his email to be removed from my web site. I will gladly do that, if only he writes a serious paper on GR, or at least reply professionally. Here are two questions:

Do you believe that the "points" from the underlying manifold can be connectedonly and exclusively only by their physical content that is invariant under "active" diffeomorphisms? If your answer is 'no', what could be “the” base state in GR (a.k.a. the reference fluid of GR), which binds the "points" by one single unique bare UNdecidable matrix?

It is generally believed that (i) one can picture the spacetime in GR as a manifold that can be "locally modeled" on some fictitious flat Minkowski space, but (ii) this picture should break down at short distances of the order of the Planck length. Neither of these ideas are needed, however. There is no need for any limitation in the possible accuracy of localization of spacetime events either. Why is that? Because the so-called Planck length may possess an inner geometrical structure.

"The problem of the definition of the concept of singularity in General Relativity is very difficult indeed, as can be appreciated by reading on its historical development (Hawking and Ellis, 1973; Tipler, Clarke and Ellis, 1980). The intuitive ideas are clear: if any physical or geometrical quantity blows up, this signals a singularity. However, there are problems of two kinds:

• the singular points, by definition, do not belong to the space-time which is only constituted by regular points. Therefore, one cannot say, in principle, “when” or “where” is the singularity.

• characterizing the singularities is also difficult, because the divergences (say) of the curvature tensor can depend on a bad choice of basis, and even if one uses only curvature invariants, independent of the bases, it can happen that all of them vanish and still there are singularities.

"The second point is a genuine property of Lorentzian geometry, that is, of the existence of one axis of time of a different nature to the space axes....

"All in all, it seems reasonable to diagnose the existence of singularities whenever there are particles (be them real or hypothetical) which go to, or respectively come from, them and disappear unexpectedly or, respectively, subito come to existence.

"And this is the basic definition of singularity (Geroch, 1968; Hawking and Ellis,1973), the existence of incomplete and inextensible curves. That is to say,curves which cannot be extended in a regular manner within the space-timeand do not take all possible values of their canonical parameter....

"Singularities in the above sense clearly reach, or come from, the edge of space-time. This is some kind of boundary, or margin, which is not part of the space-time but that, somehow, it is accessible from within it."

"A differentiable manifold is a topological manifold M together with a differentable structure, i.e. a way of defining differentiable functions on M. The natural way of doing this (forget about this "natural way" - D.C.) is to use the charts [X] to transfer the definition of differentiable functions from Rn to M."

I suppose if you (i) do not assume upfront some space-like Killing fields, (ii) don't confine your analysis to some hypothetical vacuum spacetimes, and (iii) wish to avoid poetry [Ref. 1], you may have to start from scratch,

As of April 2010, “over 2250 papers with the words ‘dark energy’ in the title have appeared on the archives since 1998, and nearly 1750 with the words ‘cosmological constant’ have appeared” (Shinji Tsujikawa, arXiv:1004.1493v1, p. 39).

I believe Michal Chodorowski explained, in arXiv:astro-ph/0610590v3, the meaning of "more and more space appears",

It is totally unclear to me how one could define an isolated system in GR, in which "more and more space appears", to start thinking about some new positive-mass conjecture and the total energy in such "isolated system". Perhaps if one can demonstrate that _no_ asymptotically flat spacetime can be stable under the conditions of such (accelerated or not) "flux" of "more space", we will face a paradoxical situation similar to the ultraviolet catastrophe of late 19th century, after which some bright mathematician will sort out
this whole mess.

If you and/or some of your colleagues know such person, please pass her/him the link above. The task is highly non-trivial and strictly mathematical, and also requires guidance from Shao Yong.

I also believe the future of new energy sources belongs to your country, but if you wish to extract energy from 3-D space, you should always follow the advice of Shao Yong. Since I haven't heard from you so far, it seems the latter turned out to be too difficult for you.

Yours sincerely,

Dimi Chakalov

-------Examine the objects as they are and you will see their true nature; look at them from your own ego and you will see only your feelings; because nature is neutral, while your feelings are only prejudice and obscurity.

Shao Yong, 1011-1077

Fooling around with alternative current is juts a waste of time.Nobody will use it, ever.

You consider Haag's 1957 idea "of interpreting the spatial extend of a measuring device and the duration of its activation as an observable localized in the corresponding spacetime region fulfilling Einstein causality and an appropriately formulated causal propagation was (and still is) metaphoric if not to say naive" (arXiv:1107.1374v3 [math-ph], p. 8). His latest views are summarizer in [Ref. 1].

There is no such thing as "local covariance principle" (arXiv:0912.2886v4 [math-ph], Sec. 6 and p. 32).

Regarding the nonobservance of the holistic aspects of QFT (arXiv:1107.1374v3 [math-ph]), see a quasi-local approach to GR & QM at

"In simple cases the event may be regarded as the interaction process between a particle and a detector. But the notion of ‘particle’ does not correspond to that of an object existing in any ontological sense. It relates to the simplest type of global state and describes possibilities, not facts. The notion of ‘partial state’ demands in addition that we ignore all possible events outside some chosen region and thus ignore possible correlations with outside events."

I learned that the Spezialisierungsmodul offers "guidance to
scientific research on open problems in mathematical physics and
general relativity", which I hope to discuss with you during the ESI
Workshop in December this year.

Meanwhile, please check out the Ansatz for 'necessary and sufficient
conditions for spacetime', which can (hopefully) eliminate 'dieser
verdammten Quantenspringerei',

To introduce "boundaries" on the physical spacetime at all (timelike, spacelike, and null) directions, I will consider an ideal dimensionless point of 'pure geometry', and will postulate a structure of such point: a dual presentation of 'the universe as ONE', interpreted along an axis (w) as both 'an infinitesimal' (S) and 'arbitrarily large volume of 3-D space' (L). Then I will suggest topological transformations of points (called 'space inversion') in an infinite-dimensional Euclidean space to obtain a 'collapse over infinity'-reduction of this unphysical Euclidean space to 'asymptotically flat 4-D spacetime' endowed with "boundaries" placed at the dual object, 'the universe as ONE'. The spacetime of 'the universe as ONE' is called 'global mode of spacetime', with equation L = S = 1 (Eq. 2), while the equation of the Arrow of Space, generating perpetual emergence and re-creation of an asymptotically flat spacetime ('collapse over infinity'), is postulated as LS = 1 (Eq. 1), in line with Virgil's dictum Mens agitat molem or Der Geist bewegt die Materie.

I will also outline the so-called 'relative
scale principle' (RSP) aimed at removing an absolute structure of space known as 'size of objects': nobody has 'the right meter'. Relative to an observer placed at the macroscopic length scale (the "middle" of w ), objects in 3-D space would look like "large" in the direction toward L and "small" in the opposite direction toward S , while a co-moving observer will not notice any chance in her 'one meter and one second', and will always experience the same "speed of light". Since 'space' is interpreted as an emergent phenomenon along the Arrow of Space, I will introduce finite templates for 'size of objects', and will argue that their scale-dependent alteration resolves the paradox of the (accelerated) "expansion" of space toward L and the (non-accelerated) "shrinking" of space toward S , as seen by an observer placed at the macroscopic length scale (the "middle" of w ), while their local alteration recovers the correct geometrical manifestation of gravity (not "curvature"). Hence one can eliminate all "dark" effects of gravity such as "black holes", "cold dark matter", and "dynamical dark energy", and amend Einstein's General Relativity with the "total field of as yet unknown structure" from the Arrow of Space.

The quest for Finite Infinity has a long history, starting from Gunnar Nordström (Über die Möglichkeit, das elektromagnetische Feld und das Gravitationsfeld zu vereiningen, Phys. Z. 15 (1914) 504-506). It is an age-old problem of General Relativity. Nothing could be more important than understanding the topological manifold of the Universe, and its dynamics.

Traditionally, experts in GR start with what I hope to derive at the end of this project: asymptotically flat spacetime with quasi-local positive mass. For example, Rick Schoen would presuppose a "smooth manifold", which has already been equipped with a "Lorentz signature metric", "asymptotic flatness", and "appropriate falloff" conditions, and then ask question like: 'why do we see positive mass only' ? A short answer: because we have an Arrow of Space. The detailed answer requires careful analysis of all initial presumptions in Rick Schoen's talk and in present-day GR textbooks.

A rigorous definition of 'isolated gravitating system', which would ensure an "asymptotic regime such that all gravitational effects are localized inside of it" (Adam Helfer) and proper boundary conditions, is still an unresolved task: there are no physically motivated boundary conditions in the case of the Einstein equations; for example, "we do not know how to build a mirror for gravitational waves" (A. Rendall), nor can we resolve the paradox of geodesic incompleteness and "black holes". Moreover, how can we define an 'isolated gravitating system' and its (obviously) positive mass if the space itself is endowed with a new, dynamical "dark" energy? The calculating machinery of ADM, suggested half a century ago, doesn't work anymore. Enter the Finite Infinity (FI).

In a nutshell, the idea of FI is to suggest a mechanism for obtaining a finite volume of Archimedean 3-D space. The very notion of 'finite 3-D space' implies the existence of two distinguishable volumes of 3-D space, separated by a "trapped" surface (cf. lion's cage below), such that we can always define the notions of 'inside vs outside' (hence "large" vs "small") and 'left vs right'. (In order to eliminate the absolute structure of 'size', we will introduce later the so-called Relative
Scale Principle, RSP.) Now, how can we introduce some process and mechanism by which 'finite space' can be fixed at all length scales, in such way that 'physical space' will never actually reach zero nor infinity but will always remain 'finite' ? We will use an infinite-dimensional Euclidean space and will introduce smooth sphere-torus conversions in it, along a new axis w , such that these sphere-torus conversions snap and fix all Archimedean volumes of 3-D space from both "below" and "above" (cf. Fig. 5). Hence space becomes both "infinite" (local mode of spacetime) and bounded from S and L (global mode of spacetime). The latter is physically unobservable, because we're stuck in the Archimedean 3-D space and don't have access to 'actual infinity'. Only Chuck Norris has been there (twice).

As in a good crime novel, all will become crystal clear at the end. Well, eventually.

Some history. The notion of 'finite infinity' (Fi) was suggested by George F R Ellis in 1984; please see:

"This led me some years ago to ask the question: ‘How far away is an effective ‘infinity’ to use in discussing boundary conditions for local physical systems of this kind?’ (...) Then incoming and outgoing radiation conditions can be imposed on that surface F, rather than at infinity or conformal infinity I as is usual [57]. (...) Furthermore the famous positive mass theorems [64] should also be generalized to this case...."This may also be the best setting for numerical calculations for ‘isolated systems’, which often talk about ‘integrating to infinity’, but in most cases do nothing of the sort. As in the rest of theoretical physics, it would be advantageous to have a theoretical framework that corresponds more closely to actual calculations - namely an integration to a surface at a finite distance from the centre of coordinates. It is usual to make that surface a null surface; the suggestion here is that it would be better to make it timelike, corresponding to the region in the real universe where the exterior is physically separated from the local system.

"So the obvious proposal [54] is that we should put boundary conditions on all fields at that distance, rather than at infinity itself, leading to the concept of a 'finite infinity' FI ... "--[54] Ellis G F R, Relativistic Cosmology: Its Nature, Aims and Problems, in: General Relativity and Gravitation, Ed. B. Bertotti et al., Reidel, Dordrecht, 1984, pp. 215-288; cf. Sec. 5.2 and Fig. 11(c).

"We cannot imagine that the stars go on forever. It’s unthinkable. But we equally cannot imagine that they come to a stop and that beyond them is nothing, and then more nothing. Unending nothing is as incomprehensible as unending something."

In order to resolve the conundrum of 'ending something', notice that the nature of the local Archimedean mode of spacetime (cf. below) is determined by 'potential infinity': every step toward the Finite Infinity provides the necessary and sufficient condition for the next step, just as in the Thompson's lamp paradox. If we go in the "direction" toward 'the infinitesimal' (S), one can (with some luck) work out a cutoff and end-point, but it will belong, again and always, to the local ("colored") mode of spacetime (see the story about 'John's jackets' below).

If we wish to work out a proper Finite Infinity, the obvious choice is to introduce a new 'cutoff and end-point' conjugated with the infinitesimal, S , and to employ the two forms of 'infinity': potential and actual. The latter is 'already completed', in the sense that (i) it does not entail any "dynamics" that can be recorded with a physical clock, and (ii) applies only to the global non-Archimedean mode of spacetime.

We begin with postulating an uncountably infinite "number" of points packed in any finite -- arbitrarily "large" or arbitrarily "small" -- line segment, plane, or volume of space in the local (physical) mode of spacetime, in such way that 'there is nothing]between[ these points'. Then we will "insert" the global mode of spacetime ]between[ the points from the local mode, and will make sure that the global mode is non-existent in the local mode, with the sole exception of the instant 'now' from the Arrow of Space, in which the two modes of spacetime coincide (cf. Fig. 1 below).

Notice the crucial difference between the two modes of spacetime: in the local mode, a test particle equipped with a clock reading its proper time will need a finite Archimedean time interval to pass through a finite Archimedean volume of space, even if the test particle travels with the speed of light. It cannot pass through a finite volume of space for 'zero time', even if the volume of space is 'tending asymptotically toward zero', as in the case of the empty set (R). Thus, it will always need 'more Archimedean time' to pass through alluncountably infinite points from the perfectly smooth continuum of the local (physical) mode of spacetime. This is the essence of 'potential infinity', and it is valid only for the local mode of spacetime.

Hence the interpretation of 'the infinitesimal' as the empty set (R): in the local mode of spacetime, it is and will always remain a finite entity, tending asymptotically toward zero, being comprised from uncountably infinite points. In the global mode, however, it is a purely geometrical, dimensionless point -- just one single geometrical point (Fig. 1).

NB: This one single geometrical point cannot be reach from/within the local (physical) mode of spacetime, for any finite duration of time, as read with a physical clock. Why? Because its physical content (jacket) is UNdecidable, after the Thompson Lamp paradox.

Thus, the empty set (R) is endowed with a structure, as hinted in Fig. 1 below, and with 'space inversion' (Fig. 1.1). It is the flashmob for the two modes of spacetime, at which they "coincide". It is also the instant 'now' (now-at-a-distance) from the Arrow of Space. And thanks to the so-called "speed of light",it has zero duration and zero spatial extension.

Compared to the local mode, the global mode of spacetime is ontologically different: it is a non-Archimedean realm of 'potential reality' (GPIs), which can traverse any finite -- arbitrarily "large" or arbitrarily "small" -- line segment, plane, or volume of space from the local (physical) mode of spacetime for zero time, as it would have been read by the clock attached to a test particle from the local mode. Namely, the global mode is equipped with 'actual/completed infinity': it can traverse the uncountably infinite points of any finite object for zero time (as it would have been read by the clock attached to the test particle from the local mode), and endows the local mode with a web of "instantaneous" correlations (as they would have been read by an inanimate clock from the local mode). With the Arrow of Space, such kinematics gives rise to a new dynamics resembling a living organism: a 'school of fish'-type bootstrapping of physical systems in their quantum and gravitational regimes, and quasi-local geodesics, in line with the rule 'think globally act locally' (hence one can introduce a background-free, relational reality and Machian quantum gravity).

The question of what kind of 'global time' runs in the global mode of spacetime, equipped with "instantaneous" actual infinity, can be answered by explaining its corresponding 'global space' and the non-Archimedean structure of the purely geometricalGPIs in it.

In general, the quest for amending Finite Infinity with an Arrow of Space is highly non-trivial, because we should also consider the following tasks:

1. The two modes of spacetime are separated by the fleeting instant 'now' from the Arrow of Space, namely, the global mode is placed in the realm of 'potential reality' (GPIs residing in the potential future in the Arrow of Space), while the local mode refers to the irreversible world of explicated GPIs in terms of facts, placed in the steadily increasing 'irreversible past' from the Arrow of Space. Such 'ever-increasing past' is what makes an 'arrow', due to irreversible 'information gain': every instant 'now' pertains to an explicated physical universe, which contains more information that its immediate predecessor in the 'ever increasing past', and at the same instant 'now' the physical universe is offered an enriched spectrum of potential states to choose from for its next instant 'now', just as in the cognitive cycle of Ulric Neisser (Fig. 2 and Ch. 2 and 4).

Metaphorically speaking, the Arrow of Space is depicted with the Dragon devouring its tail (Ouroboros), from the Chrysopoeia ('Gold Making') of Cleopatra during the Alexandrian Period in Egypt. The enclosed words mean 'The All is One.'

The ultimate source for such information gain is 'the true monad without windows' which remains at absolute restwithin the instant 'now' (see below), depicted with Fig. 1 below.

1.1. The only "meeting point" of the two modes of spacetime is the instant 'now', in which they coincide (Fig. 1). The result is an already completed and already negotiatedphysical universe, spanned across the absolute instant 'now' from the Arrow of Space, with 3-D space and zero "thickness" along w (cf. below), in which “there’s energy in the gravitational field, but it’s negative, so it exactly cancels the energy you think is being gained in the matter fields” (Sean Carroll); check out the Photoshop layers metaphor below and the resulting non-linear dynamics here.

2. The unique "boundaries" on 3-D space in all (timelike, null, and spacelike) directions, introduced with Finite Infinity, should make the universe an 'isolated system', which (i) contains 'absolutely everything', included its Aristotelian First Cause, (ii) is self-enclosed (Albert Einstein), and (iii) is "bounded" by some ambient unphysical spacetime (called here 'global mode of spacetime') which is part and parcel from the same 'isolated system'.

2.1. To describe such 'universe as ONE' -- self-wrapped (cf. (iii) above) and endowed with the faculty of 'self-acting' due to its "gravitational field" -- one needs to place its source "inside" the instant 'now' from the Arrow of Space (Fig. 1), and make sure that the so-called "dark" energy of 'the universe as ONE' is unobservable in principle.

3. Last but not least, the ultimate source of 'the universe as ONE' should be interpreted as 'zero nothing', that is, a special kind of "zero" opposite to the mundane case of 'zero something'. Perhaps the only way we could grasp the notion of 'zero nothing' or 'the true monad without windows' is by recalling the relation between the content and volume of concepts: the larger the volume, the smaller the content; hence the source of 'absolutely everything, the unknown unknown included' should possess zero intrinsic content (cf. the undefinable matrix).

I haven't been able to find suitable mathematical formalism to describe these widely known ideas. All I can offer is a simple (but incomplete) geometrical explanation of Finite Infinity.

Firstly, there should exist a maximal volume of 3-D space (L), at which we place the Finite Infinity (FI), such that any finite volume of space, no matter how large, is identified as an Archimedean sub-volume smaller than the 'maximal space volume', L. Likewise, there should exist a minimal volume of 3-D space (S), at which we place the same Finite Infinity (FI), such that any finite volume of space, no matter how small, is identified as an Archimedean volume larger than 'the minimal space volume', S .

Secondly, the two physical, finite, Archimedean volumes of space tend asymptotically toward L and S in the local (physical) mode of spacetime, but cannot reach them for any finite Archimedean duration of time due to the "structure" of the empty set (R) at which the global mode and the local mode coincide (Fig. 1).

The 'no-go' axiom about the empty set (R) ("cannot reach them for anyArchimedean duration of time") stems from our belief (not 'fact') that there exist, in the local mode of spacetime, an 'uncountably infinite' "number" of points between any finite Archimedean volumes of space and their "two" cutoffs, Land S: the empty set (R) cannot be bridged by any finite Archimedean system for any finite duration of time, even if it jumps over the points with the "speed of light". That is, in the local mode of spacetime the empty set (R) is 'finite' entity, and will always occupy a finite, albeit "very small", volume of space packed with 'uncountably infinite' points arranged by 'potential infinity'. On the other hand, the same empty set (R) is consumed/traversed instantaneously in the global mode of spacetime, which has non-Archimedean nature and is endowed with 'actual/completed' infinity.

A precise explanation of 'physically reach': in the local (physical) mode of spacetime, we always reach/nullify the empty set (R) at the point of '2 min', which is why the state of Thompson's lamp is always defined by the rules of 'bartenders', but it is just a fleeting "jacket" cast by John from the global mode of spacetime. The latter is endowed with actual infinity. In other words, in the global mode of spacetime the interval [0, 2] is closed, while in the local mode the same interval is open -- (0, 2).

Thanks to the Arrow of Space, the 'large finite volume' will chase L in the local mode indefinitely; in the local mode of spacetime the empty set (R) can only tend asymptotically toward zero. The same holds for the opposite case of the separation of any arbitrarily small Archimedean volume of space from S by the same empty set (R): in the local mode, it would require an infinite -- actual infinity -- amount of time for the elimination of the empty set (R) and reducing it to zero, at which point the Arrow of Space would stop, and the universe would cease to exist.

The very notion of 'space' requires that all physical objects acquire finiteArchimedean size; a table with length 'one meter' fills in a template for 'one meter'. In the case of an one-meter template, we observe a finite table with length one meter, which is again a 'sub-volume of finite space'. The purpose of Finite Infinity (FI) is to define the largest templateand the smallest template for all finite volumes of 3-D space.

The maximal volume of 3-D space, L , and minimal volume of 3-D space, S , belong to the global mode of spacetime for which the actual infinity holds ('already completed'), and are related by

LS = 1 (Eq. 1).

Thanks to the empty set (R), no physical, Archimedean stuff can 'physically reach' L and hence go "beyond" L , and also no physical stuff can 'physically reach'S and hence go "below" S (compared it to the conformal recipe): L & Sis ONE object which belongs to the non-Archimedeanglobal mode of spacetime.

In a nutshell, the modified Finite Infinity satisfied two conditions: in the local mode of spacetime the universe approaches asymptotically its causal boundary (cf. Eq. 1 above) within an open interval, while at the same time the universe is being (present continuous) permanently wrapped by itself in the global mode of spacetime. Hence 3-D space can be literally wrapped by itself, exactly as Albert Einstein required.

Again, there is no path whatsoever to the global mode of spacetime from the local mode of spacetime; hence the FI Postulate above. The difference between S and L is that in the case of the former we can find some 'numerically finite but physically unattainable boundary/cut off' (the Planck length), but for L we have only an eternally expanding'sub-volume of finite space', which can only chase L but can never physically reach it: L and S belong to the non-Archimedeanglobal mode of spacetime.

It is the Arrow of Space which creates such asymmetry in the treatment of L and S . We have a numerically finite but physically unattainable "bottom", while space "expands" in the local mode toward Lindefinitely: at each and every instant 'now' from the Arrow of Space, the universe occupies a finite sub-volume, being literally wrapped from both "below" and "above" with Eq. 1 above. Depending on the direction we look at the edge of the universe in the local mode of spacetime, either toward L or toward S, we see "two" edges, while in the global mode they are ONE -- the universe as ONE, as stressed by Lucretius some 2060 years ago. We don't have such structure of spacetime in differential geometry textbooks.

Notice that we face here a kind of 'logical infinity': no physical, Archimedean volume of space can reach L , because L & S belong to the 'the universe as ONE' (Eq. 1 above). In other words, one could logically reach something only if this "something" is 'not yet reached', while in our case it is logically impossible to "reach" something (L) which is being eternally residing "inside" us (S) from the outset. Thus, the notion of 'logical infinity' refers to the statement that if the Arrow of Space is terminated, the universe will cease to exist and will return to its initial state of pure light and cognition, known as [John 1:1].

Without the global mode of spacetime, the universe would be governed exclusively by Archimedean geometry, conformal recipes would have worked, tessellation of space with 3-D "tiles" would have been possible, and we would calculate the exact finite number of "atoms" filling a finite volume of space without any gaps from the empty set (R): we would hit an "end-point" beyond which "is nothing, and then more nothing" (S. Leacock). Thank God, this is impossible.Again, we cannot physically "see" L & S, because we don't have access to 'actual infinity'. Only Chuck Norris has been there (twice).----------

Now, let me try to explain geometrically the empty set (R), which would "point" to the "dimension" of space, denoted with w (from wunderbar, after Theodor Kaluza), of the postulated global mode of spacetime (pictured with red; local mode with black).

The size of physical bodies along warezero, because w lives only at the fleeting instant 'now' -- a purely geometrical "point" at which the two modes of spacetime, global and local, coincide. Attached to the same instant 'now' is a 'pocket of propensity states' in the global mode of spacetime, called General Platonic Ideas (GPIs); check out GPIs in the human brain (not "mind") here.

First, the dimensionless instant 'now', at which the two modes of spacetime coincide:

The so-called 'space symmetry' in the global mode of spacetime (pictured with red) is defined as two interchangeable andsimultaneous presentations of any finite volume of space from the local mode (pictured with black circle), such that a globalobserver can monitor anyfinite volume of space in the twodirections of wsimultaneously, as explained in Wiki: (i) from thecenter of the circle (Fig. 1) "outside" in all radial directions, and(ii) from "outside" the black circle (2-D trapped surface of lion's cage) toward the center of the circle along the same T-inverted radial directions (not shown). In 2-D space, the "trapped surface" is a circle; in 3-D space it is a sphere with radius r (Fig. 4.2). In order to derive the circumference of the black circle, consider the two red rectangles (sides n = 4), and start doubling their parameter n , after Archimedes. At the instant 'now' from the Arrow of Space (Fig. 1), n reaches actual/completed infinity and the inscribed and the circumscribed polygons shift to 'pure geometry' of 'the grin of the cat without the cat', as observed by Alice (in the LHS of Einstein field equation).

The sides of the polygons are converted into uncountably infinite "geometrical points", each of which can take a fleeting physical (colored) "jacket" in the local mode of spacetime (cf. the parable of John's jackets below).

If we apply space inversion along the two "directions" in the drawing from Mark Armstrong below, we can grasp the notion of Finite Infinity: every volume of 3-D space is being snapped like the circumference of a circle, along the two "directions" of w (see Fig. 5 below), thanks to which it becomes a finite entity. Voila!

In the local mode of spacetime (pictured with black), we multiply the dimensionless point in Fig. 1 along one dimension in two opposite directions, to obtain 2-D spacetime. For the purpose of this presentation, the time direction will not be shown. Notice that 1-D space is endowed with T-invariance: we can flip all the points to the right with those to the left (Fig. 2.1), and vice versa, with respect to the initial point in the middle (Fig. 2.2).

Fig. 2.1

Fig. 2.2

The next step is constructing 2-D space from Fig. 2.2, by introducing a second spatial dimension, again with two directional degrees of freedom, and move all points from 1-D space, en bloc, in two opposite directions (Fig. 3.1).

Fig. 3.1

Fig. 3.2

Fig. 3.3

Notice that Fig. 3.2 is a Flatland universe (time dimension not shown) in which Flatlanders enjoy PT-invariance: swapping the points along the horizontal line (Fig. 3.2), with respect to the vertical line, will bring T-symmetry, while the vertical flip, with respect to the horizontal line, will produce a mirror image with 'right' and 'left' interchanged. If we try to apply Finite Infinity to such Flatland universe, the two spatial dimensions will be "bounded" by the non-physical (to Flatlanders) third spatial dimension in which their space would look extrinsically "curved", with "tangential surfaces" attached to one "point" but spanned in the third spatial dimension.

In the current GR textbooks and tutorials, the black and red points in Fig. 3.3 are fused with the rules of diff calculus, after Leibniz; then people claim that in "sufficiently small" (notice the poetry) neighborhood around such fused black/red point "it is possible to choose a "locally inertial coordinate system" such that, within a sufficiently small region of the point in question, the laws of nature take the same form as in unaccelerated Cartesian coordinate systems in the absence of gravitation" (cf. below). What a mess.

Going back to the Flatland: we, as "global" observers, can "see" all points from Fig. 3.2 en bloc, while the Flatlanders will encounter obstructions from a line: it will be like an infinite 1-D wall which prevents them from seeing "behind" the wall. We can "see" all the points from Fig. 3.2 simultaneously, en bloc, which would translate to Flatlanders as 'keeping simultaneously two inverted images from their Flatland', one with T-symmetry and a second one with P-symmetry. Regarding their 'time', the Flatlanders will be totally puzzled by our ability to oversee en bloc their two paths from A to B in their (t+)-direction, and from B to A in their (t-)-direction. They don't have such clock, and will claim that our "time" is dead frozen (much like 3-D people do in canonical quantum gravity).

It requires far more efforts to elaborate on the example from Wiki about an infinite 2-D plane in our 3-D space, which poses no restrictions to "observers" in the global mode along w , equipped with actual infinity and capable to "see" simultaneously all points from our 3-D space both en bloc and inverted with 'space inversion' (cf. Fig. 5). Notice that we have three such restrictive planes in 3-D space (xy, xz, and yz, cf. Fig. 4.2), and enjoy CTP-invariance upon "swapping" the points via the global mode of spacetime, as we did in Figs 2.1 and 3.1 above.

The new symmetry, called 'space inversion', should eliminate all fixed relations in the Archimedean 3-D space, such as 'large vs small' or 'inside vs outside'. Otherwise our 3-D space will not be dynamical, but will impose a fixed background for the length scale in terms of absolute size of objects in 3-D space. Once we eliminate the absolute Newtonian time, we should eliminate all absolute structures from 3-D space as well.

To elaborate on the example from Wiki, I suppose one could "see", from the global mode of spacetime, two superposed P-invariant images of "the lion", obtained after the inverted (w.r.t. the cage surface) 3-D space. Such symmetry is literally about inverting all the "points" trapped inside a finite volume of space approaching asymptotically Swith all the "points" from the finite volume of space approaching asymptotically, in "opposite direction", L , which remain outside that "shrinking" finite volume of space (Fig. 1.1). The "number" of points in any finite volume of space is 'uncountably infinite', so there should be no obstacles to perform such 'space inversion'.

Let's start with endowing the Flatland with a spatial dimension, by moving all the points from Fig. 3.2 en bloc along two opposite "directions", up (toward L) and down (toward S):

The two opposite "directions" along w , inferred from Fig. 4.1, are absent in Fig. 4.2. One is toward the Small (S); the other runs in the opposite direction along w , toward the Large (L). In the instant 'now' (Fig. 1), these additional geometrical degrees of freedom are set to zero, as explained in the beginning of this exercise -- they are hidden withinthe empty set (R). The meaning of 'set to zero' corresponds to 'spontaneous broken symmetry' (cf. below): we end up with only one "charge" of mass, and the wunderbar (after Theodor Kaluza) dimension w is being literally eliminated, thanks to which we obtain the good old 3-D space of classical physics. It doesn't contain any trace from the "negative mass", but two worlds with inverted spacetime basis, material and tachyonic, separated by a luxonic "surface":

Max Tegmark, arXiv:gr-qc/9702052v2, Footnote 4: "The only remaining possibility is the rather contrived case where data is specified on a null hypersurface. To measure such data, an observer would need to "live on the light cone", i.e., travel with the speed of light, which means that it would subjectively not perceive any time at all (its proper time would stand still)."

Due to the 'spontaneous broken symmetry' (cf. below), we have two kinds of mass in the picture above, real and imaginary (tachyonic), safely separated by a timeless luxonic world of zero, or rather indifferent mass, which keeps the dimension wtotally hidden.

Notice that, in addition to the three planes in 3-D space (xy, xz, and yz) and their corresponding transformations or "swapping the points" (CPT-invariance), we have a new 'space inversion' symmetry, as mentioned in the discussion of Fig. 1.1 above.

NB: This exercise will be really tough. Try to imagine the "trapped circle" from Fig. 1.1 as "trapped 3-D space", that is, a 3-D "rubber glove" hypersurface which can be seen along the two directions of w as two simultaneous "gloves", right and left, corresponding to the unbroken symmetry of the GPI field (see below) inhabited by the unphysical, GPI states of negative & positive mass.

You will have the unique freedom to "look" at the 3-D hypersurface simultaneously along the two directions of w : from 'inside-out' and from 'outside-inside' (cf. the drawing from Mark Armstrong below), corresponding to 'sphere' and 'torus' (cf. Fig. 5).

Such simultaneous viewpoint is impossible to imagine, as hinted in Wiki, but if we lower the dimensions of the physical space to 2-D spherical surface trapped by the six red planes in Fig. 4.2, the 'inside-out' view will display all points from the 2-D spherical surface, as seen from the center of Fig. 4.2 along all radial directions 'inside-out', and by executing 'space inversion' with respect to the 2-D spherical surface ("lion's cage") we will see again all points from the finite 2-D surface from 'outside-inside', along the inverted radial directions, keeping the two presentations of the 2-D "rubber glove" (left and right) simultaneously available to our w - inspection. However, we cannot "move" the unique object at the center of Fig. 4.2, shown with the red point in Fig. 1 above: it is in 'absolute rest' and is residing both inside every point from the physical 3-D space and outside the physical 3-D space, as The Beginning (see below) of the two viewpoints at the 3-D "rubber glove" hypersurface along w , from 'inside-out' and from 'outside-inside'.

Last but not least, the aim of the putative space inversion is to recover 'asymptotically flat spacetime' with the two 'viewpoints' along w . Look at Fig. 3.2 above, and picture it as a clock, such that Fig. 1 is placed at the center, and four red segments tangential to 12, 3, 6, and 9. Imagine one radius "connecting" (not quite: see Fig. 4.3) the center to 9, and blow up the circle by instructing the radius to reach infinity and pass "over" it. The conventional wisdom tells us that the circle will "degenerate", or rather "collapse", to 1-D Euclidean space presented with the vertical red line at 9 (Fig. 3.2) at the very instant at which its radius is exactly infinite, but at that same instant the two points in the orthogonal direction, 12 and 6, will ultimately break up. Once it passes "over" infinity, the object will regain its 2-D status, but will be converted into two finite cross-sections of a torus. Notice that at the instant of 'collapse over infinity' all points from the circle in Fig. 3.2 will be arranged in 1-D space, like those in Fig. 2.2 above.

Now, if we keep the radius constant, such non-smooth topological transition will match a well-known screensaver in Windows 97, half of which is depicted with the drawing from D. DeCarlo and D. Metaxas (1996) below.

But our case is different, since we wish to recover the asymptotically flat spacetime "around" the critical 'collapse over infinity' instant, with positive and negative space curvature inherited from the circle and the torus. Namely, after passing through this critical instant, not just one but all "inflating" radial directions -- except for the center, see Fig. 1 -- will be inverted, as explained in the exercise above. If we wish to start by inflating the torus from 'case (e)' above, along the opposite direction in w , we will again pass through the same 'collapse over infinity' instant, "around" which we imagine some 'asymptotically flat spacetime', after which we will wind up in the "clock" case from Fig. 3.2, only this time its radius will be shrinking toward the center of the clock. Please keep these two invertible T-invariant images passing through 'asymptotically flat spacetime' in your mind. All we need now is to replace the circle/torus with the 3-D "rubber glove" hypersurface from the exercise above.

Again, the two opposite directions along w are totally hidden inside the luxonic world (cf. Max Tegmark above) with indifferent (zero) mass. Physically, the two directions of the circle/torus conversion in the global mode of spacetime, along w , aretotally hidden, their duration is zero, and we can only imagine two atemporal waves, depicted with the Escher hands, by which all non-linear negotiations in 'the school of fish' are completed for zero time, as read with your wristwatch: we can never see Macavity.

Notice that I haven't tried so far to introduce any 'distance function'. I have only an uncountably-infinite 'points' ordered with the relation of intermediacy, [A (zero) B], which is interpreted as "zero is ]between[ A and B". The unique object with zero physical presence in the local (physical) mode of spacetime is The Aristotelian Connection of 'the universe as ONE' (cf. the red dot in Fig. 1). It is manifested by purely affine connection facilitating parallel transport of vectors in the local (physical) mode of spacetime, from one spacetime point to the nearest point, and is independent from the metric. It also captures the essence of the "speed" of light. Let me try to explain.

0.47-0.52: "Relative to the platform, time on the train completely stops."The ill-defined expression " √0 " must be amended by a new mathematical object relevant exclusively to the non-Archimedean global mode of spacetime.

Relative to the passengers in the train (see the picture above), the local mode of their 'time on the train' does indeed "stop": you move into the luxonic world (cf. Max Tegmark above), and can "see" all the intermediate topological transformations of the Universe, not just its 'collapse over infinity'-state depicted in Fig. 5.

Consider again a finite chunk of space, say, one meter: it is comprised from uncountably-infinite 'points' ordered with the relation of intermediacy, [A (zero) B]. The Aristotelian Connection of 'the universe as ONE', [A (zero) B], passes the physical state at point A to its "neighboring" point B with particular "speed" which acquires an upper limit for any finite volume of space. The numerical value of the "speed of light" isn't interesting; just like the Planck length, it is a 'numerically finite but physically unattainable boundary of the local mode of spacetime'. We cannot define, even as a Gedankenexperiment, 'one second' as the product of [Planck time] x 1043. The universe as ONE (global mode) is a totally different world which shows up only with its Aristotelian Connection.

What matters here is that The Aristotelian Connection operates in the global mode of spacetime: it "reads" all of the uncountably-infinite 'points' from any finite chunk of space en bloc . By an analogy with a movie reel, one could speculate that The Aristotelian Connection is "projecting" 3-D space as 'uncountably-infinite infinitesimal "frames" per second', meaning that every "frame" is the 'collapse over infinity'-state (cf. Fig. 5). Unless we encounter relativistic effects (watch the movie from NOVA below), this en bloc"reading" does not slow down (for comparison, watch a water drop here).

The unique "speed" of light, due to The Aristotelian Connection, is preventing us from taking even a glimpse at the global mode of spacetime: we can see only an already-completed physical world, in line with the idea about relativistic causality ("causal processes or signals can propagate only within the light-cone," Jeremy Butterfield).

Notice the precise meaning in GR of 'already-completed': all non-linear negotiations between the two sides of the Einstein field equations are being post factum completed for zero time, as read with your wristwatch. "All agree that in general relativity, the metric tensor gij is (or better: represents a field that is) dynamical: it acts and is (at the same instant - D.C.) acted on. They also agree that it is a special field since it couples to every other one, and also cannot vanish anywhere in spacetime. Many authors go on to say that the metric tensor represents geometry, or spacetime structure, so that geometry or spacetime structure acts and is (at the same instant - D.C.)acted on" (Jeremy Butterfield). More succinctly: "the metric is treated as a field which not only affects, but also is (at the same instant - D.C.) affected by, the other fields" (John Baez). This non-linear paradox is depicted with the Escher hands above, and can be resolved only with the two modes of spacetime, as argued previously. Michael Redhead argued in 1995 that the notion of 'localizable particle' makes sense only for a free particle, while I suggest a 'quasi-local' alternative in terms of Machian relational ontology with the rule 'with respect to everything else in the universe'. In short, the issue of relativistic causality is anything but trivial in GR. As stressed by Margaret Hawton, in quantum field theory probability density is defined at a fixed instantt , and it is by no means obvious how to "insert" in such fixed instantt the already-completed non-linear 'acting and at the same instant being acted upon' in GR.

Thanks to The Aristotelian Connection [A (zero) B], there exists a bond between spacetime points, and the "speed" of light is its manifestation. If the "speed" were infinite or unlimited, there would be no difference between 'small' and 'large', and '3-D space' as the medium for 'finite things' would have not existed; if the "speed" of light were a finite entity, it would be surpassable, and then there would be no separation between the two worlds with inverted spacetime basis, material and tachyonic (cf. Max Tegmark above).

Notice that 'space as the medium for finite things' is a very old idea, debated by Ernst Mach. The difference between 'small' and 'large' is not fixed in the local, Archimedean mode of spacetime, but in the global, non-Archimedean mode by 'finite invariant templates' (cf. below).

NB: The alteration of these templates is the essence of gravity. We have 'finite invariant templates' for 'finite space', fixed by The Aristotelian Connection: it is the bond between spacetime points that determines 'the time it would take a photon to traverse a distance equal to a [finite volume of space]'. (As Erik Curiel explained, arXiv:0908.3322v3 [gr-qc], "in general relativity all the fundamental units one uses to define stress-energy, namely time, length and mass, can themselves be defined using only the unit of time; these are so-called geometrized units. (...) A unit of length is then defined as that in which light travels in vacuo in one time-unit.") Then a minuscule alteration of the bond between spacetime points can produce enormous change of 'the time it would take a photon', and subsequently of the 'finite invariant templates' (resembling "curvature", cf. Bill Unruh), leaving the deceitful impression of some "cold dark matter" or "supermassive black hole".

The effect is purely geometrical, according to the so-called relative scale principle (RSP). Namely, an observer placed inside such altered (by 'the time it would take a photon') spacetime template will not notice any change of her ambient spacetime assembled by The Aristotelian Connection, in terms of her 'one meter and one second': she will always experience the same "speed of light", regardless of the extent to which her template has been altered with respect to an unaltered template of an observer placed at the length scale of tables and chairs. This is because the "number" of spacetime points in any finite volume of 3-D space, approaching asymptotically S and L , remains unchanged (Kurt Gödel), being a non-Archimedeanphenomenon, and also because The Aristotelian Connection is endowed with 'actual infinity' and "reads" alluncountably-infinite 'points' from any finite chunk of space en bloc .

Notice that 'the time it would take a photon', which defines each and every 'finite volume of space' (see above), is the global (en bloc) mode of time. It is the background time code of the animation from John Walker below, and it must be totally hidden.

There is
a lot more to be said about the "speed" of light, as The Aristotelian Connection "happens" only post-factum, at null-surface. We cannot witness the alteration of these templates online, as it "happens", and we will always observe an already-assembled, by The Aristotelian Connection, spacetime in which we cannot reach the luxonic world.

By going into the center of Milky Way, all we can post-factum observe is a finite volume of space in which our 'one meter and one second' has not been altered, hence we would claim that these 'spacetime templates' are "invariant to us", while a distant observer will see us "shrinking". Ditto to the "expansion" of space. There is no need to invoke any anomalous "dark" stuff with "positive energy density and negative pressure", as you may have heard from Ned Wright, say.

More on SPR later; for now it suffices to say that the "speed" of light must be 'numerically finite but physically unattainable boundary' in order to provide for finite volumes of space and time intervals of the local mode of spacetime.

First and foremost, I need to eliminate all absolute structures in 3-D space, such as 'absolute size', by allowing the 'templates for finite space' to shrink toward S and expand toward L , as viewed by an observer at the length scale of tables and chairs, and also providing invariant size of all object toward S and toward L , as seen by a co-moving observer. In other words, we need a dual, scale-dependent metric to define 'distances in 3-D space'. Then the "expansion of space" can be explained as an illusion observed only from the length scale of tables and chairs. Ditto to the "non-accelerated" shrinking of the "size" of objects toward S , in line with the so-called 'relative
scale principle' (RSP). The latter is an elaboration on the old idea about 'the mutual penetration of the Large and the Small'. Who has "the right meter"? Nobody. This is the essence of
RSP.Simple, no?

NB: To explain the 'templates for finite space' and their "dark" influence on matter and fields, recall the operational definition of 'second': the total duration of 9,192,631,770 transitions between the two hyperfine levels of the ground state of the cesium 133 atom, provided the cesium atom is "at rest at a temperature of 0K, such that the ground state is defined at zero magnetic field" (reference here). In the local, Archimedean mode of spacetime, we imagine that the distance between every two neighboring dots from the drawing below refers to one transition between the two levels of the ground state of the cesium-133 atom, and that the total of such physical, Archimedean distances, comprising 'one second', is exactly 9,192,631,700:

{..............................................}

This is clearly an unfeasible Gedankenexperiment, which cannot be reproduced. Most importantly, the "intrinsic time interval associated to any timelike displacement" (Ted Jacobson, pp. 18-19), defined here as 'the elementary tick of time ]between[ two purely geometrical points ordered with the relation of intermediacy [A (zero) B]', can be defined only in the global, non-Archimedean mode of spacetime. These are 'invariant templates' for finite durations of time and finite volumes of space, which approach asymptoticallyL and S .

NB: An objection to such 'finite templates' would be that they are merely mental, or epistemological constructs representing our "knowledge", hence we cannot grant them an ontological status of 'ideal geometrical reality'. Well, with Finite Infinity we can indeed suggest such 'finite templates' fixed by the topological transitions in the global mode of spacetime (cf. Fig. 5), hence resolve the underlying puzzle of 'extendable volumes of space', which allow us to "look around, and see as far as we can" (L. Smolin). This simple fact can be explained iff the spacetime "points" are perfect geometrical points ordered by The Aristotelian Connection as a perfect continuum (the current theory requires that "points become fuzzy and locality looses any precise meaning," cf. Sergio Doplicher).

The alteration of these templates, relative to an observer at macroscopic length scale, produces purely geometrical effects (interpreted as "black holes" and "expanding space"), yet a co-moving observer will not notice any alteration of her intrinsic 'distance function'. Again, this is the well-known idea about 'the mutual penetration of the Large and the Small', which can start only from the macroscopic length scale in the two "directions" along w , toward L and toward S .

With
RSP and its dual spacetime metric, one could alter the "distances" in 3-D space in such a way that a "reactionless" alien visiting craft (AVC) will travel with speed "one meter per second" (relative to the people in the AVC) to cover the size of a galaxy (relative to observers outside the AVC). You won't notice your "speeding", but you will notice that the space in front of you is running towards you, just like with the Alcubierre warp drive. In RSP, the question of 'who has the right meter and the right second' is meaningless. In present-day GR, the question of 'who has the right meter' is answered with absolute size of objects in 3-D space, toward L and toward S , which is nothing but 'looking for a right answer to a wrong question'.

Fig. 4.3 below shows the main idea of Finite Infinity.

Fig. 4.3The horizontal line shows the axis w in terms of two directions in 3-D space, toward S and L . The points denoted with
B (from ‘bartenders’) are the fleeting explications of GPIs in the local (physical) mode of spacetime: John’sjackets.

And finally, let's see how the w axis will look in the global mode of spacetime (Fig. 5), by combining Fig. 1 with the blow up of Fig. 3.2 at the point " 9 o'clock " (not shown).

Fig. 5

The red/black point in Fig. 5 belongs to an asymptotically flat, physical, 4-D spacetime; the vertical black line is taken from Fig. 2.2. above. The right part from Fig. 5 shows the T-invariant (cf. the drawing from Mark Armstrong above) "blow up" of the circle from Fig. 3.2, by watching the point at "9 o'clock" (see above): at the critical 'collapse over actual infinity' instant, at which the radius of the circle in Fig. 3.2 is exactly infinite, we would have obtained an absolutely flat 4-D spacetime, which is why I talked, for the lack of better wording, about "around the critical 'collapse over infinity' instant". The left part from Fig. 5 shows a segment from a torus -- see the drawing from DeCarlo and Metaxas above, 'case (e)', and the explanation of 'space inversion', about the two simultaneous viewpoints at the 3-D "rubber glove" Cauchy hypersurface, from 'inside-out' (right part from Fig. 5) and from 'outside-inside' (left part from Fig. 5). The horizontal w axis is the one along which 'the mutual penetration of the Small and the Large' begins from the macroscopic length scale, as discussed above. Hence the only remnant from w in our asymptotically flat spacetime (called 'local mode of spacetime') is depicted with the two red lines in Fig. 4.3 above, placed at S and at L .

Notice that the horizontal w axis is not a 4th spatial dimension, because the new degrees of freedom to "look at" the 3-D "rubber glove" Cauchy hypersurface entail moving simultaneously along all directions in the local mode of spacetime, from any point in 3-D space, from both 'inside-out' toward L and its time-reverted direction toward S . This is impossible in 3-D space, as explained in Wiki below, but recall that these are the "directions" of the Arrow of Space: the elementary 'change of space' is nothing but the elementary increment of our physical time, as read with our clocks -- it isn't a "vector". Thus, the global mode of spacetime and the axis w of the Arrow of Space could only be perceived by us as "occurring" in some infinite-dimensional Euclidean space, much like a Flatlander (cf. Fig. 3.2) would be totally puzzled by our 3-D viewpoint, and would also have to imagine some higher-dimensional space to accommodate our 3-D viewpoints inside his Flatland.

To visualize these smooth torus-sphere transitions over the 'collapse over infinity' instant 'now' (Fig. 1), hence the re-created 'asymptotically flat 3-D space', look carefully at the drawing below, from Eric Schechter (5 December 2009, emphasis added): "There are no points for plus or minus infinity on the line, but it is natural to attach those "numbers" to the endpoints of the semicircle."

The point denoted with 'zero' from the vertical line corresponds to the point at "9 o'clock" in Fig. 3.2 above (not shown). Notice that every point from the circle in Fig. 3.2 above will pass over the 'collapse over infinity'-instant, and at this instant its two conjugated endpoints,from the direction orthogonal to the horizontal line segment pictured here, will "break up" the circle and will convert it into a (hyper?) torus. The 3-D space at the exact 'collapse over infinity' instant would be absolutely flat, and would contain just a bare red point from Fig. 1 above, known as [John 1:1]. Which is why I talked, for the lack of better wording, about "around the critical 'collapse over infinity' instant", to describe the asymptotically flat 3-D space, hence 4-D spacetime, of present-day GR.

But look at Eq. 1 above: in the global mode of spacetime, we can set L and S to take values of some dimensionless variable "measured" along w ; all we need is to ensure that L and S take reciprocal values, until they snap to

L = S = 1 (Eq. 2).

Eq. 2 describes 'the whole universe as ONE' in its global mode of spacetime, inhabited only by GPIs. The Arrow of Space runs simultaneously along the two "opposite directions" from w , and at each and every instant 'now' (see Fig. 1) a newly-born physical universe is being re-created in the local mode of spacetime.

Perhaps the global mode of spacetime can be presented with four segments (notice the favicon of my web site) in which two atemporal quantum-gravitational waves run against each other, re-creating asymptotically flat spacetime at the 'collapse over infinity' instant. Also, the fundamental phenomenon called "spin" should be explained as topological property of 3-D space. As Peter Rowlands suggested (arXiv:0912.3433v1, Sec. 3, p. 5), “Space and time are simply quaternions multiplied by i, and spin is simply a topological property of space (as Dirac knew), and not quantum or relativistic in origin.”

In the quantum-and-gravitational realm of the local (physical) mode of spacetime, physical objects gradually acquire increasing access to 'the whole universe as ONE'. They become bootstrapped by their "gravitational field", due to opening a "red window" to their GPIs residing in the global mode of spacetime. Which is why a quantum particle can be in a superposition of its GPI states (say, a superposition of |cat> and |dog>, Erich Joos), and all (not just "small") bodies follow quasi-localgeodesics: at each point from such quasi-local geodesics we can install a local Lorentzian frame, by going into "freefall" at such quasi-local point, and imagine that the effects of gravity have re-disappeared.

Hundreds of millions U.S. Dollars and Euro -- taxpayers' money -- have been wasted so far by LIGO Scientific Collaboration, and even more are scheduled to be wasted with the "enhanced" and "advanced" LIGO and LISA.

I seriously urge you and your colleagues to examine your problems professionally. Please feel free to pass this email to all participants of your 8th International LISA Symposium.

NB: Should you or any of your colleagues have *professional* questions, please do write me back. Notice I will not reply to insults nor to emotional statements.

Your recent paper [Ref. 1] is a joy to read, and also a tough challenge to study. I haven't yet completed the second part, but since you acknowledged that Problem C (determining the gravitational correlation) is unresolved, may I offer some thoughts on the subject matter.

It seems to me that many physicists are unaware of the fundamental puzzle in differential calculus, as shown with the Thompson Lamp paradox,

They write textbooks and teach GR like a bartender [Ref. 2]. Other "bartenders" suggest various ad hoc solutions to the "dark" constituents of the universe in the following fashion:

Q: What is green, lives underground, has one eye, and eats stones?A: The One-Eyed Green Underground Stone Eating Monster!

I believe both dark matter and dark energy are artefacts of our essentially incomplete presentation of 'the infinitesimal', which may in turn be resolved with some pre-geometric plenum "connecting" x' and x [Ref. 1] dynamically, along an 'arrow of spacetime',

p. 1: "Can there be an alternative description for these observational effects that does not assume the existence of these mysterious dark quantities?

"One possibility is that both dark matter and dark energy are artefacts of some effective averaged theory of gravitation.....Problem A (How does one Average tensor fields on a manifold?)Problem C (What is the nature of the gravitational Correlation , C_ab?)....p. 7: "3.2. Choice 1: Parallel Transport along Geodesic

"To begin, we must first select a unique curve that connects the points x and x' and a connection: for our purposes, we choose the geodesic and the Levi-Cevita connection. The geodesic is a “natural” choice as there are no other “natural” curves that connect x' and x. In Riemannian space, the geodesic is the shortest and straightest path connecting points x' and x. A weakness in this approach is the assumption that a unique geodesic exists connecting x' and x.....p. 8: "We have illustrated a covariant averaging procedure for tensor fields addressing problem A. We have not averaged the Einstein Field Equation’s of General Relativity, and therefore have not addressed problem C of determining the gravitational correlation, so much more work to do."

An infinite crowd of mathematicians enters a bar. The first one orders a pint, the second one a half pint, the third one a quarter pint... "I understand", says the bartender - and pours two pints.

[Ref. 3] Masumi Kasai (23 June 2009): "May the Force be with the inhomogeneous cosmologists. May the Force be with us."IPMU International Conference dark energy: lighting up the darkness!June 22-26, 2009, http://web.ipmu.jp/seminar/darkenergy09/MKasai.pdf

"It is remarkable that even classical dynamics of gravity asks for dimension > 4. As two and three dimensions were not big enough for free propagation of gravity, similarly four dimension is not big enough to fully accommodate self interaction dynamics of gravity. Then the most pertinent question is where does this chain end?"

It ends at infinitely-dimensional spacetime -- see Fig. 2 below. Every infinitesimal "point" from the local mode is endowed with infinitely-many connections (global mode of spacetime) with 'the rest of points' in the local mode. And since the global mode is hidden by the so-called 'speed of light', the
bootstrapped local mode is "self-acting" upon itself along the arrow of spacetime. Not surprisingly, the origin of this "self-force" is not traceable, and some people consider it "dark".
This offers a new interpretation of the old idea of "breathing" (inhaling/exhaling) universe (common knowledge in India), only the duration of the "breathing" cycle is exactly zero in the local mode of spacetime, rendering the latter a perfect continuum. That's the proposal for 'quantum principle for spacetime dynamics', after
Schrödinger and KS Theorem.

Perhaps Naresh Dadhich would some day accidentally discover it. All he has to do is to forget about "branes" and other multidimensional superstitious.

If this just a hobby? For if you were collecting stamps, while I was suggesting to switch to collecting paper napkins, I could understand your attitude of neglecting the underlying "raw potato" -- the UNdecidable KS state,

NB: Category theory cannot -- not even in principle -- model the "raw potato". You and John are wasting your time and FQXi donations: $89,981 for your efforts, and $131,865 for John's "Categorifying Fundamental Physics".

I read with great interest your arXiv:0705.2908v2, but couldn't understand the following:

"When the positive and negative matters with the same mass meet, they will become a real vacuum." And also: "The positive and negative matters under some exceeding conditions may be created from nothing at the same time. They will also be main tests of the existence of negative matter."

What could be the ultimate source of positive and negative matters (called "nothing"), such that it can "meet" them as "a real vacuum"?

You said (Thu, 23 Dec 2004) that the affine structure "is a further primitive (not definable from mere differential structure) structure which you can postulate using some representation or other of it" -- please see

I meant something pretty simple by what I wrote in Shape of Space. Consider a space which has the structure only of a differential manifold. Then, so far, no affine structure, no geodesics, no curvature, no Christoffel tensor. The transition from this to affine structure is not given by or extruded from Christoffel symbols or the 3-tensors which they represent. The affine structure is a further primitive (not definable from mere differential structure) structure which you can postulate using some representation or other of it. You can postulate it as a covariant derivative, a connection, or a tensor which can be represented in coordinates by a Christoffel symbol. But that representation makes sense only if the affine structure is already there, so to speak. True, in GR, the fundamental equationtells us (among other things) that the curvature and the "matter distribution" are co-determinate. That doesn't mean that the curvature is caused by the matter tensor. A simple analogy shows the catch in that way of thinking. The distance relations between London, New York and Sydney entail that the cities aren't on a flat surface. But the distances don't cause the shape of the surface. These places couldn't have those distances if the surface wasn't curved in the first place. The basic equation of GR places a mutual constraint on the tensors oneach side of it.

I guess you know that the tensor as represented by a Christoffel symbol isn't straightforwardly like other tensors. If you don't, B. Schutz A First Course in General Relativity sec. 5.5 gives a clear account of it.

I will be on vacation until Sept. 8 and will have no email contact formost of that time. If you cannot wait until I return, please contactDenise Henry (dshenry@nsf.gov) .

-----------

Note: At the age of 57, my hair is getting lily-white, and I may claim that it is sufficiently white to enjoy surfing at porno in the web, yet still sufficiently black to attract real blond girls. Likewise, LSC members deeply believe that GW strain, which might hit LIGO some day, would be sufficiently weak to be modeled with the linearized approximation of GR, yet sufficiently strong to be detected with the so-called Advanced LIGO (cf. J. G. Pereira et al., arXiv:0909.4408v1).

But LSC can't have their cake and eat it. As Hermann Weyl demonstrated in 1944 (Hermann Weyl, How Far Can One Get With a Linear Field Theory of Gravitation in Flat Space-Time? American Journal of Mathematics, Vol. 66, No. 4 (Oct., 1944), pp. 591-604), the linearized approximation of GR is "a shadow without power".

Notice that LSC cannot describe smooth bi-directional transitions from strong GWs to very weak GWs, as they approach asymptotically their sudden death at the stage of "a shadow without power". Joshua Goldberg is manifestly silent on this crucial problem, and Kip Thorne didn't even mention Hermann Weyl's article in his lecture "Gravitational waves in flat spacetime". Instead, he tried to defend the so-called "invariance angle" of LIGO's arms with an article by Eugene Winger, which he has read as a student "around 1960". Just look at the L-shaped tunnels of LIGO: isn't this 'graviton parapsychology'?

The persistent "evolution" of the beliefs of LSC members is really amazing. Back in 1981 (cf. Daniel Kennefick, p. 1), Kip Thorne had no difficulty in "finding a taker for a wager that gravitational waves would be detected by the end of the last century. The wager was made with the astronomer Jeremiah Ostriker, one of the better-known critics of the large detectors then being proposed. Thorne was one of the chief movers behind the largest of the new detector projects, the half-billion-dollar Laser Interferometer Gravitational Wave Observatory, or LIGO. He lost the bet, of course."

Now LSC members are effectively saying 'just gives us a couple of billion dollars more, and we gonna make it'. Exactly how much more? The Advanced LIGO Cost Estimating Plan (M990310-05.pdf, updated 05.27.2003) is here, but is hidden to 'mass society' taxpayers.

"It is worth reminding ourselves why and where GR might fail." (...) "Any such failure of GR should point the way to new physics."

Once the "Advanced LIGO" fails in 2015, they will celebrate the 100th anniversary of Einstein's GR with their "way to new physics", claiming that their total failure is actually of fundamental importance, like the negative result for the ether drift in the experiment of Michelson and Morley ... but with just a few billion dollars more for LISA and Einstein Telescope, everything will be just right.

Are NSF officials going to risk a devastating embarrassment from their blind support of GW parapsychology?

So, even if we assume that Kip Thorne has somehow missed the monograph by V. Denisov and A. Logunov, he is most certainly aware that the whole "GW astronomy" is in murky waters since the discovery of "dark" energy in 1997 by Adam Riess,

"It is worth reminding ourselves why and where GR might fail." (...) "Any such failure of GR should point the way to new physics."

Once the "Advanced LIGO" fails in 2015, they will celebrate the 100th anniversary of Einstein's GR with their "way to new physics", claiming that their total failure is actually of fundamental importance, like the negative result for the ether drift in the experiment of Michelson and Morley ... but with just a few billion dollars more for LISA and Einstein Telescope, everything will be just right.

I am afraid NSF officials are indeed taking the risk of devastating embarrassment after their blind support of GW parapsychology. And they will get it:

Perhaps you can help your US colleagues. You know the work by Viktor Denisov, Anatol Logunov, and Ludvig Faddeev. I hope you have some spare time to write up a brief paper and post it on arxiv.org server, to prevent the abuse of Einstein's GR with the 'Plan B' above.

Nobody should blame GR for the forthcoming failures to detect GWs with some "enhanced" or "advanced" LIGO. Einstein's errors regarding energy transport by GWs were identified even before Kip Thorne was born. I will be happy to provide you the references.

Looking forward to hearing from you,

Yours sincerely,

Dimi Chakalov-------------

Note: The staggering problem of "GW astronomy", encoded in the asymptotic expression r --> [infinity] , can perhaps be explained by an ancient Greek in the following fashion. Suppose Achilles is throwing his famous spear in one direction along the radius of the universe, r , starting from its center at Athens. He isn't familiar with the wisdom of present-day cosmology, but is a staunch relativist and knows that the universe is like an unbroken ring with no circumference, for the circumference is nowhere, and the "center" is everywhere.

How can Achilles prove the 'no circumference' conjecture? Suppose that, at some advanced stage from his exercise, he finds out that cannot throw his spear anymore, because he has reached some (asymptotic) limit of being 'too far away from Athens'. Namely, the increasing distance (r) between Achilles and Athens somehow blocks his spear at r --> [infinity] , placing the circumference of the universe at some "effective infinity" from all "centers", Athens included. But because Achilles is smart relativist, he will be immediately puzzled by the privileged (if not absolute) location of Athens in the universe and its influence on his spear, such that he cannot replace Athens with some closer point from his path (say, just one point behind the last location of his spear) and throw his spear further, ad infinitum.

This is very unclear, isn't it? As Adam Helfer put it (arXiv:0903.3016v1 [gr-qc]), the asymptotic spacelike regime is "less well understood mathematically at present". English translation: it's pure poetry, as stated above.

The situation with the alleged null infinity isn't better either: "From a physical point of view, null infinity is very far away." (Bernard F. Schutz, Mathematical and Physical Perspectives on Gravitational Radiation, August 2, 2002)

To paraphrase Woody Allen, infinity is very long, especially towards the end. People tend to indulge themselves with some "rescaling metric" recipe (Ted Newman), but it is totally unclear how to "rescale" the metric during its accelerated expansion, as driven by the "dark" energy from empty space.
In this gloomy situation, may I offer some optimistic speculations. John Stachel mentioned a seminal paper by Niels Bohr and Leon Rosenfeld, Zur Frage der Messbarkeit der elektromagnetischen Feldgrssen, published in 1933: Because EM charges "occur with two signs that can neutralize each other, a charge-current distribution acting as a source of an electromagnetic field can be manipulated by matter that is electrically neutral and so not acting as a source of a further electromagnetic field; and one can shield against the effects of a charge-current distribution." Then he added: "A glance at Bohr and Rosenfeld 1933 shows how important the possibility of neutralizing the charges on test bodies is for measurement of the (averaged) components of the electric field with arbitrary accuracy, for example. This difference may well have important implications for the measurement of gravitational field quantities."

So, we need some entity that is charge-neutral to the two signs of mass: potential reality. Also, because GWs are sheer coordinate effects, they might "propagate" with any velocity desired by the human imagination, included 'the speed of thought' (Arthur Eddington, The Propagation of Gravitational Waves, 1922). We have to stick to the full non-linear GR, because its linearized approximation is "a shadow without power" (Hermann Weyl; see above).

All this has been said many times at this web site; sorry for repeating it here. The only correction I need to make concerns my statement above: "Einstein's errors regarding energy transport by GWs were identified even before Kip Thorne was born." But I was wrong. Hermann Weyl's article was published in 1944, at the time when Kip Stephen Thorne was 4 year old. Sorry, I was wrong. Mea culpa.

Please note CEOFOP's claim that "a true, real stress-energy-momentum tensor for gravity does also exist!", provided "all of the metric is handled as "dynamical" " [Ref. 1]. The result is that "the energy in gravity and that in matter always balances out to zero" (ibid).

This reminds me of Merced Montesinos' paper [Ref. 2] on 'the right answer to the wrong question' (MTW p. 467).

NB: I wonder how you would comment on CEOFOP's "true, real stress-energy-momentum tensor for gravity", given his statement that "Einstein's equations are non-linear, and this is why gravitational fields can be the source of additional amount of gravity, so that a gravitational field can support itself." [Ref. 1]

Surely we don't see "the ether" coming back to GR textbooks [Ref. 2], but would you please explain your viewpoint on the above-mentioned "true, real stress-energy-momentum tensor for gravity" ?

Please also check out CEOFOP's interpretation of the "radial coordinate r " used in the Schwarzschild (actually, it is Hilbert-Droste-Weyl) solution [Ref. 1], and compare it with the interpretation offered by Angelo Loinger [Ref. 3].

"What does L say about this? "I have proven that dynamical solutions do not exist, so your solution is wrong". What is wrong about it? First, he ignores the wave packets and focuses on the plane wave solutions. These have infinite extension in space and time and represent infinite energy. That, indeed, is problematic in gravity. If the energy in a given region with linear dimensions R exceeds R in natural units, a black hole is formed so that space-time undergoes asubtle change in topology. This might arguably be called unacceptable. The problem is manifest in our explicit solutions, and this is why it is important to use wave packets instead. The wave packages are identical to the ones in Maxwell theory, and since they represent only finite amounts of energy (per unit of length in the z direction), these solutions are indeed legitimate. I showed L how to construct explicit, analytical examples of such wave packets.".....

"Actually, a true, real stress-energy-momentum tensor for gravity does also exist! If all of the metric is handled as "dynamical" one finds that the Einstein tensor G_µv itself acts as the gravitational part of the energy-momentum. Adding this to the energy and momentum of matter one finds a quantity that is trivially conserved: the addition gives zero, according to Einstein's equations. Thus, the energy in gravity and that in matter always balances out to zero. In practice, this is not a very useful definition; it would imply that gravity carries a gigantic amount of energy, most of which is invisible, and no further physical information is obtained, but it is the matter of principle (Sic! - D.C.) that counts here. In practice, we may wish to ignore the large contribution from the background, and this is why a "pseudotensor" emerges. One can add to this that, by construction, the pseudotensor should only depend on first derivatives, whereas the "true tensor" G_µv contains second derivatives, which makes it physically counter intuitive.".........

" "Black holes do not exist; they are solutions of the equation for the Ricci tensor R_µv = 0, so they cannot carry any mass. And what is usually called a "horizon" is actually a physical singularity."

"Mr. C. adds more claims to this: In our modern notation, a radial coordinate r is used to describe the Schwarzschild solution, the prototype of a black hole. "That's not a radial distance!", he shouts. "To get the radial distance you have to integrate the square root of the radial component grr of the metric!!" Now that happens to be right, but a non-issue; in practice we use r just because it is a more convenient coordinate, and every astrophysicist knows that an accuratecalculation of the radial distance, if needed, would be obtained by doing exactly that integral. "r is defined by the inverse of the Gaussian curvature", C continues, but this happens to be true only for the spherically symmetric case. For the Kerr and Kerr-Newman metric, this is no longer true. Moreover, the Gaussian curvature is not locally measurable so a bad definition indeed for a radial coordinate. And why should one need such a definition? We have invariance under coordinate transformations. If so desired, we can use any coordinate we like. The Kruskal-Szekeres coordinates are an example. The Finkelstein coordinates another. Look at the many different ways one can map the surface of the Earth on a flat surface. Is one mapping more fundamental than another?

"The horizon is a real singularity because at that spot the metric signature switches from (+,-,-,-) to (-,+,-,-)", C continues. This is wrong. The switch takes place when the usual Schwarzschild coordinates are used, but does not imply any singularity. The switch disappears in coordinates that are regular at the horizon, such as the Kruskal-Szekeres coordinates. That's why there is no physical singularity at the horizon.

"But where does the black hole mass come from? Where is the source of this mass? R_µv = 0 seems to imply that there is no matter at all, and yet the thing has mass! Here, both L and C suffer from the misconception that a gravitational field cannot have a mass of its own. But Einstein's equations are non-linear, and this is why gravitational fields can be the source of additional amount of gravity, so that a gravitational field can support itself. In particle theories (Sic! - D.C.), similar things can happen if fields obey non-linear equations, we call these solutions "solitons". A black hole looks like a soliton, but actually it is a bit more complicated than that.

"The truth is that gravitational energy plus material energy together obey the
energy conservation law. And now there is a thing that L and C fail to grasp: a black hole can be seen to be formed when matter implodes. Start with a regular, spherically symmetric (or approximately spherically symmetric) configuration of matter, such as a heavy star or a star cluster. Assume that it obeys an equation of state. If, according to this equation of state, the pressure stays sufficiently low, one can calculate that this ball of matter will contract under its own weight. The calculation is not hard and has been carried out many times; indeed, it is a useful exercise for students. According to Einstein's equations, the contraction continues until the pressure is sufficiently high to stop any further contraction. If that pressure is not high enough, the contraction continues and the result is well-known: a black hole forms. Matter travels onwards to the singularity at r = 0, and becomes invisible to the outside observer. All this is elementary exercise, and not in doubt by any serious researcher."

[Ref. 2] Merced Montesinos, The double role of Einstein's equations: as equations of motion and as vanishing energy-momentum tensor, arXiv:gr-qc/0311001v1, http://arxiv.org/abs/gr-qc/0311001, pp. 4-5.

"This means that for this type of observers, there is a balance between the 'content' of energy and momentum densities and stress associated with the matter fields [psi] (which is characterized in Tµv) and the 'content' of energy and momentum densities and stress associated with the gravitational field (which is characterized in [XXX])

--->--->--->---><---<---<---<--- (23)

in a precise form such that both fluxes cancel, and thus leading to a vanishing 'flux', i.e., tµv = 0. Once again, the vanishing property of tµv for the system of gravity coupled to matter fields is just a reflection of the fact that the background metric is dynamical.

"More precisely, tµv = 0 tells us that the 'reaction' of the dynamical background metric is such that it just cancels the effect of 'flux' associated with the matter fields. It is impossible (and makes no sense) to have a locally non-vanishing 'flux' in this situation. If this were the case, there would be no explanation for the origin of that non-vanishing 'flux'. Moreover, that hypothetic non-vanishing 'flux' would define privileged observers associated with it (the ether would come back!).

"It is important to emphasize that, in the case of having a dynamical background metric, the vanishing property of tµv = 0 is not interpreted here as a ‘problem’ that must be corrected somehow but exactly the other way around. In our opinion, there is nothing wrong with that property because it just reflects the double role that the equations of motion associated with the dynamical background play."

There is no need to invent the wheel: check out Schrodinger at the link.

As I said six and a half years ago ("energy-momentum flows from matterto grav. fields and back", see below), you can bring a horse to the water, but cannot make him drink.

Dimi----

On Thu, 11 Mar 2004 08:17:00 +0100, "Hooft 't G." wrote:>> Let me briefly explain. Following the conventional Einstein> equations, the matter-energy-momentum tensor is COVARIANTLY> conserved; it is not conserved if you replace covariant derivatives> by ordinary derivatives. This is how energy-momentum flows from> matter to grav. fields and back.-----

Note: Gerard 't Hooft has been generously "casting pearls to the swine" (exact quote -- see CEOFOP_1.pdf in CEOFOP_1.zip) at

Check out a snapshot (2.11.2010) from the page above, Gerardus_energy.jpg, and notice the splitting of "the metric gμν into a background part, goμν, for which we could take flat space-time, and a dynamical part: substitute in the Einstein-Hilbert action: gμν = goμν + g1μν . (...) Just require that the background metric goμν obeys the gravitational equations itself; one can then remove from the Lagrangian all terms linear in g1μν. This way, one gets an action that starts out with terms quadratic in g1μν, while all its indices are connected through the background field goμν."

It is utter madness indeed. Notice that this person is Chief Editor of Foundations of Physics (CEOFOP), and maybe (hope not!) teaches GR. That's really scary.

As to the latest note by Gerard 't Hooft, entitled "The plane gravitational wave for beginners" (Addendum 18/8/2010), he failed to mention that, for a pp-wave, all curvature invariants vanish (Hans Stephani and John Stewart, General Relativity, Cambridge University Press, 1982; section 15.3). That's 'for beginners'; I've said much more in ExplanatoryNote.pdf. Just a hint: the proper calculation of 'the self force' is not "miniscule" but shows the input of "dark energy" and the mechanism by which GWs transport energy and momentum; as Hermann Bondi stressed many years ago, "the question of the "reality" of the waves essentially concerned whether they transported energy. Such transport is a fundamentally nonlinear phenomenon." LIGO is manifestly blind and deaf to this inherently nonlinear phenomenon, and cannot measure anything related to
BMS group.

2010/3/18 Dimi Chakalov <dchakalov@gmail.com>:>> Dear colleagues,>> If you wish to be removed from this thread, please say so.>> The whole issue about GR is very simple indeed; I managed to explain> it even to my teenage daughter,>> http://www.god-does-not-play-dice.net/#Bahn>> All the best,>> Dimi>

Secondly, you grounded your paper on the assumption that Einstein's quadrupole radiation formula might be correct, and tried to modify the filed equations instead (p. 425).

You wrote (p. 423): "In view of the fact that there is no existing gravity energy-stress tensor, it seems simple and natural to assume that the source tensor T_ab is zero in a vacuum."

Then you argued (ibid.) that "Einstein's radiation formula implies that his field equation must be modified so that the source tensor is nonzero in vacuum."

Why not have it both ways? Yes you can: T_ab (the energy momentum stress tensor of all matter and fields) can *completely* vanish/dissolve into the vacuum, and stay available there for any partial, full, or "over unity" recall, if and when needed. All you need is a new form of reality: see my note on the dynamics of GR at

> Let's not forget that Einstein's pseudo-tensor is a meaningless concoction of> mathematical symbols because it implies, by contraction, a linear> invariant that depends solely upon the components of the metric tensor> and their first derivatives. But G. Ricci-Curbastro and T. Levi-Civita> proved in 1900 that such invariants do not exist. Mr. 't Hoof does not> understand this.

The problem is that he is Chief Editor of Foundations of Physics, and also teaches GR. He can do a lot of damage, mostly to his students.

> Furthermore, 't Hooft concedes that the total energy> of Einstein's gravitational field is always zero. This implies that> the the field equations violate the usual conservation of energy and> momentum so well-established by experiment.

"A third player, DC, strongly supports L and C, and on the side asks me to seriously consider his theories about the 9-11 events: the two planes crashing into the Twin Towers have first been snatched by UFO's, their passengers were abducted, and the planes, without passengers and filled with explosives of an unknown type, were directed into the towers. All of this to explain why the towers collapsed in spite of their impeccable design. I can only try to guess who came up first with this theory, but I now use it to illustrate the level of my discussions with DC. "

To explain the level of my discussions with CEOFOP, I invite you to check out the facts at my 9/11 web page,

Note: The only truth in CEOFOP's statements above is that I have indeed asked him (along with Prof. Brian Josephson) to seriously consider my theory about the 9/11 events: see my email from Fri, 25 Dec 2009 printed below.

All the rest -- "two planes crashing into the Twin Towers have first been snatched by UFO's, their passengers were abducted, and the planes, without passengers and filled with explosives of an unknown type, were directed into the towers" -- is untrue.

Why did CEOFOP (G. 't Hooft) write all this crap at his web site, I wonder. He is fluent in English, so we have two alternatives. One explanation could be that he has somehow lost his intellect, but very selectively, only regarding my 9/11 web page. Another explanation would be that he had actually understood the simple text at my web page, but decided to lie about it, for unknown (to me) reasons.

In simple terms: he is either a selective moron, or just a bold liar. But not both.

I will leave the decision to his students in GR. If they come up with a third option, I will immediately post it here.

> Mr. 't Hooft uses the linearised form of the field equations. He is> evidently ignorant of the fact that Hermann Weyl proved, in 1944, that> linearisation is inadmissible because it implies the existence of a tensor> that, except for the trivial case of being precisely zero, does not> otherwise exist.

> Concerning the quantity 'r' in the "Schwarzschild solution", Mr. 't Hooft,> in his lecture notes, calls it the shortest distance to the centre. He also> calls it the radial coordinate or coordinate radius. He even told me once> that it is a gauge choice that defines what 'r' is. It has never been> correctly identified by any proponent of the black hole nonsense. All these> concepts are false because it is irrefutably the inverse square root of the> Gaussian curvature of the spherically symmetric geodesic surface in the> spatial section and is thereby not even a distance in the related manifold.> Here is my detailed analysis of this:>> http://www.ptep-online.com/index_files/2007/PP-09-14.PDF>> http://www.ptep-online.com/index_files/2008/PP-12-11.PDF

"One way to see how this works, is to split the metric g_mv into a background part, [X], for which we could take flat space-time, and a dynamical part: [XX]......."The stress-energy-momentum tensor can then be obtained routinely by considering infinitesimal variations of the background part, just like one does for any other type of matter field; the infinitesimal change of the total action (the space-time integral of the Lagrange density) then yields the stress-energy-momentum tensor. Of course, one finds that the dynamical part of the metric indeed carries energy and momentum, just as one expects in a gravitational field. As hydro-electric plants and the daily tides show, there's lots of energy in gravity, and this agrees perfectly with Einstein's original equations."

"Any doubts about these facts are removed once the existence and properties of the Green functions for the linearized theory have been established. These Green functions can then be used to study systematic expansions to obtain the solutions of the complete, non-linear theory, to any required accuracy. Good theoretical physicists completely control the proper use of Green functions. (...) I did construct them, and found that, provided due attention is paid to the gauge freedom in the use of coordinates, these functions are well-behaved."

It is "casting pearls to the swine", says CEOFOP (Chief Editor Of Foundations Of Physics).

This is not a joke: check out the link above. The PDF file from his web page is available, too.

Final quote: "A third player, DC, strongly supports L and C, but his claims are too opaque for me to even address."

To get Gerard ’t Hooft's "pearls" of wisdom, all you need is to split the metric gmv into a "background part" (to obtain "flat space-time"), and a dynamical part that "carries energy and momentum", as "hydro-electric plants and the daily tides show". Just don't forget to use "well-behaved" Green functions.

A colleague of mine offered only a brief comment: "It's madness, utter madness."

He claims (May 12, 2010) that "theories without any Higgs particle are possible but ugly and have been practically ruled out by observations", but failed to comment on Howard Georgi's unparticles and my prediction from January 9, 2003.As Howard Georgi explained, “there could be a scale-invariant world separate from our own that is hidden from us”. My comment: we can "see" this scale-invariant world with our brains only, as we know since Plato.

LHC is deaf and blind to the scale-invariant world, hence will only "see" that the number of quarks is jumping to 8 and more, in a Fibonacci sequence.

It seems to me that his idea is a bona fide case of Murphy's Law No. 15. I suspect that CDM and DDE are due to spacetime acting upon itself, hence the "acting agent" cannot _in principle_ be traced back to any concrete physical stuff,

It seems to me that you are trying to explain one 'unknown' with another 'unknown'.

As you stated in [Ref. 1, p. 2], the topic of your essay -- the origin of the arrow of time -- is based on the singularity theorems of GR. Take Hawking-Penrose theorem, published in 1970. It presupposes some specific energy conditions [Refs. 2 and 3], which cannot hold in a world dominated by "dark energy"; see Rakhi and Indulekha at

On top of everything, the hypothetical gravitational "collapse" is highly controversial in the first place [Ref. 4].

Perhaps it will be a good idea if you first sort out the unsolved mysteries in the basis of your essay [Ref. 1], as neither S. Hawking nor R. Penrose were anticipating some "dark energy" in 1970s.

I will appreciate your professional reply, as well as the comments from your colleagues.

Kindest regards,

Dimi

-----------[Ref. 1] Claus Kiefer, Can the Arrow of Time be understood from Quantum Cosmology? arXiv:0910.5836v1 [gr-qc], to appear in "The Arrow of Time", ed. by L. Mersini-Houghton and R. Vaas, http://arxiv.org/abs/0910.5836

From the abstract: "Remarks are also made concerning (...) scenarios motivated by dark energy."

p. 2: "The question raised by the presence of all these arrows is whether a common master arrow of time is behind all of them...."As indicated by the singularity theorems of general relativity, a consistent description of the Big Bang may require a new framework such as quantum gravity. The question then arises whether the origin of the arrow of time can be understood there. This is the topic of my essay....p. 11: "Since our present Universe is dominated by dark energy, which for our purpose here can be approximated by a cosmological constant [lambda], ......p. 12: "In the case of a non-vanishing cosmological constant ... "

----------Note: Let's try to find out what we know that we don't know regarding 'The Arrow of Time'.

1. Claus Kiefer speculated extensively about 'entropy' [Ref. 1], but we "do not have a rigorous definition of entropy for systems coupled to gravity" [Ref. 5; see also the discussion of Eq. 1 on p. 7 therein].

2. We are also conditioned to believe that, since the Einstein field equations are "local", it would be impossible that "the global topology of the spatial section of an FLRW universe could have an effect on that universe's dynamics" [Ref. 6]. But Einstein field equations themselves are totally inadequate for addressing the main puzzle of non-unitary emergence of 'energy from empty space' -- see above. This opens an opportunity to seek 'the arrow of time' in global topology changes, to at least gather some hints as to how the Einstein field equations should be modified to tackle the source of "dynamic dark energy".

3. If we look at the "expanding" FRW universe above, the first puzzle we should notice is that the cosmological time, as read by our wristwatch, should not be observable: while FRW equations describe the physical time evolution, the "observed" quantities are "not gauge invariant and therefore should not be observable in obvious contradiction to reality", cf. T. Thiemann [Ref. 7].

3.1.Thomas Thiemann has argued that a tentative solution may be offered with some of those "scalar fields" introduced ad hoc [Ref. 7], to bypass the direct approach to the "dark energy" from the quantum vacuum. Unfortunately, all those "scalar fields" are red herring, firstly because there is no symmetry mechanism preventing their strong coupling -- see Sec. 2.2 in [Ref. 8].

p. 4: "Transition functions relabel the points that constitute M, which for the time being we think of as recognizable entities, as mathematicians do. (For physicists these points are mere ‘potential events’ and do not have an obvious individuality beyond an actual, yet unknown, event that realizes this
potentiality.)...p. 11: "An absolute structure is a coordinate which takes the same range of values in each Diff(M) orbit and therefore cannot separate any two of them. If we regard Diff(M) as a gauge group, i.e. that Diff(M)-related configurations are physically indistinguishable, then absolute structures carry
no observable content."

(ii) my email messages sent in the past three years (no reply from you), and (iii) the latest confirmation of Lieu & Hillman's direct evidence against Planck-scale fluctuations in spacetime by Jonathan Granot,

> I am truly grateful to all of you who wish to vindicate our 2003 work,> but note also that our paper was six years ago and times were> different. The Fermi limit is indeed unique, as they tested> systematic rather than random 1st order Planck scale effect. Giovanni> is right - the Fermi test was not about fluctuations, but real> dispersion.

Of course. What I did say in my preceding email was: "The common issue is the end result from two very different approaches, by Richard and Jonathan. That is what matters."

Maybe I should have added 'that is what matters to the question in the subject line'.

Regrettably, Giovanni can't see the forest for the trees. Details about 'the forest' at

"The endorsement process is not peer review. You should know the person that you endorse or you should see the paper that the person intends to submit. We don't expect you to read the paper in detail, or verify that the work is correct, but you should check that the paper is appropriate for the subject area. You should not endorse the author if the author is unfamiliar with the basic facts of the field, or if the work is entirely disconnected with current work in the area."

Stanley Deser refused to even consider endorsing my manuscript. I do hope some of his colleagues will agree.

If you, my dear reader, have papers "that have been submitted between three months and five years ago" to [gr-qc] or [astro-ph] domains, and would agree to endorse the submission of my manuscript, please contact me by email as soon as possible, and I will send you my manuscript.

Thanks for your reply. I regret that you refused to endorse the submission of my manuscript to [gr-qc]. As you put it in lrr-2007-3,

"The event horizon is a global property of an entire spacetime and is defined nonlocally in time: The event horizon in a slice is defined in terms of (and cannot be computed without knowing) the full future development of that slice."

---------Note: Forty years have passed since Roger Penrose suggested the so-called Cosmic Censorship Conjecture (CCC), and people continue to ground their efforts on it (cf. Jonathan Thornburg and Michael I. Cohen et al., p. 2) and offer bets and drawings of "quantum horizon geometry".

Notice that the methods for finding an accurate history of the so-called event horizon (you need the "event horizon" (EH) in order to speculate about "black holes" viz. "binary black hole mergers" as "one of the most astrophysically common sources of gravitational radiation for detectors such as LIGO", cf. Mike Cohen et al.) require complete knowledge of "the full future evolution of the spacetime before the EH can be determined exactly" (ibid.).

Two examples: Geroch's theorem regarding CTCs (Robert Geroch, Topology in general relativity, J. Math. Phys. 8, 782-786 (1967)) and Tipler's theorem, which stipulates that, given the (obvious) possitiveness of the energy-momentum tensor, there can be no changes in the topology of spacetime, hence the causal structure of spacetime is secured (Frank J. Tipler, Singularities and causality violation, Ann. Phys. 108, 1-36 (1977)), are from the old days of GR without DDE.

I would like to make two comments on your online article "In search of the black swans" [Ref. 1].

I'm interested in proactive management of emerging risks, type "black swans". With the benefit of hindsight, it seems to me that the event of 9/11 could not have been predicted, for reasons entire different than Max Planck's discovery; please see

The second comment pertains to the subject of this email: I believe Lee Smolin [Refs. 2 and 3], and all of his colleagues, have completely missed a fourth road to quantum gravity, from Erwin Schrödinger,

"In 1890 an electricity company enticed the German physicist Max Planck to help it in its efforts to make more efficient light bulbs. Planck, as a theorist, naturally started with the fundamentals and soon became enmeshed in the thorny problem of explaining the spectrum of black-body radiation, which he eventually did by introducing the idea — a “purely formal” assumption, as he then considered it — that electromagnetic energy can only be emitted or absorbed in discrete quanta. The rest is history. Electric light bulbs and mathematical necessity led Planck to discover quantum theory and to kick start the most significant scientific revolution of the 20th century....."... it is not the normal events, the mundane and expected “white swans” that matter the most, but the outliers, the completely unexpected “black swans”. In the context of history, think 11 September 2001 or the invention of the Web."

"Time in the Newtonian schema is a parameter used to label points on a trajectory describing the system evolving in configuration space. When the system is small and isolated, this time parameter refers to the reading of a clock on the wall of the observer’s laboratory, which is not a property of the system. When we try to apply this notion to the universe as a whole, the time parameter must disappear (see John Baez - D.C.). Some have attempted to argue that this means that time itself does not exist at a cosmological scale, but that is the wrong conclusion. What disappears is not time, but the clock outside of the system -- which would be an absurd object since the system is the whole universe."

p. 205: "One of the biggest mysteries is that we live in a world in which it is possible to look around, and see as far as we like."

p. 206: "God is nothing but the power of the universe as a whole to organize itself."

----------

Note: Nothing from the text above is original. The legacy of Erwin Schrödinger and Henry Margenau is anything but some unpredictable "black swan". For comparison, if someone has been offering the guiding principles for HTTP protocol to the people at CERN, but they were refusing to develop it, the invention of the Web [Ref. 1] would have never happened.

One can only hope that the upcoming sixth consecutive failure of LSC to detect GWs with the "Enhanced LIGO" will shake up the established theoretical physics community. If not, we will have to wait for the seventh consecutive failure of LSC in 2015, with their "Advanced LIGO".

I'm also glad you quoted Roland's monograph, "The Interpretation of Quantum Mechanics", in which he demonstrated that by introducing the decoherence functional you're automatically confined to Boolean logic -- a clear indication for Murphy's Law No. 15, "Complex problems have simple, easy-to-understand wrong answers."

Since you mentioned in your talk the case depicted on Fig 1 from your arXiv:0909.2597v1 [gr-qc], I wonder whether you would, in such kind of "space", be able to look around and see as far as you like (Lee Smolin, "Three Roads to Quantum Gravity", Phoenix, 2000, p. 205). The test of the pudding, you know.

Fay: I'm glad you chose (synchronicity?) to sit next to me. If some day you decide to talk about the topology of space (with or without its "dark energy"), please drop me a line. I'll be delighted to attend.

Lastly, may I use this opportunity to invite all of you to join the new research program outlined at

Suppose you have a clear night sky, and fix your eyes, for 1 min, on a star that is/was 4M light years away. The star looks the same during the whole time of 1 min, and is always at the place you see it.

"Central to both the classical and quantum problems is the notion of an entire trajectory. At the classical level it appears to be the appropriate reparametrization-invariant notion for the construction of interesting probabilities. At the quantum level, the decoherent histories approach appears to handle the problem in a natural way, perhaps because it readily incorporates the notion of trajectory."-----------

"What about the semiclassical approximation and the recovery of an appropriate external time parameter in some limit? (...) The discussion is also connected to the question: Where does the imaginary unit i in the (functional) Schrödinger equation come from? The full Wheeler-DeWitt equation is real, and one would thus also expect real solutions for [PSI]."-----------

Please notice that the source of DDE (see ‘GR with DDE’ at the link above) is assumed to play the role of ‘reference fluid’ and Anderson's ‘absolute structures’ (James L. Anderson, Principles of Relativity Physics, Academic Press, New York, 1967, p. 73). The latter (i) do not obey the ‘generalized principle of action and reaction’ (ibid., p. 339), and (ii) can be revealed in Quantum Theory as ‘potential reality’,

It seems to me that all the pieces of the jigsaw puzzle of quantum gravity snap to their places -- effortlessly.

Should you or any of your colleagues disagree, please bark back. The "curious incident" with my web site is that nobody is willing to comment on these very simple (and certainly not original) ideas.

Kindest regards,

Dimi Chakalov--------"Is there any other point to which you would wish to draw my attention?""To the curious incident of the dog in the night-time.""The dog did nothing in the night-time.""That was the curious incident."

I believe kids have the right to know everything we know. If you consider Nevill Mott's paper (arXiv:0911.5136v1 [math-ph], p. 18) and the alleged "time-dependent decoherence factor" (W.H. Zurek et al., arXiv:0911.5729v1 [quant-ph]), I believe your students will grasp the ultimate puzzle of our good old asymptotically flat spacetime, and will never waste their life with chasing ghosts.

Kindest regards,

Dimi Chakalov------

Note: Contrary to what you may hear from Jonathan Halliwell, it is impossible to explain the trace of a charged particle in a bubble chamber, after Nevill Mott (see S. Doplicher, p. 18), with the current QM and GR textbooks. The problems from QM (A. Teta, arXiv:0905.1467v1, "possible histories", pp. 12-14) are swept under the carpet. Moreover, the problems from 'GR without DDE' are overwhelming. One is that we can't explain 'the energy-momentum density of generic background quantum states' (see below): "points become fuzzy and locality looses any precise meaning" (S. Doplicher, p. 21). More puzzles from M. Consoli and E. Costanzo, arXiv:0709.4101, Sec. 3: Is the vacuum a preferred frame? Could be. But if we model the vacuum as 'potential reality', there is no way to observe motion with respect to it: the usual expression from 1930s -- "the only Lorentz-invariant tensor" (ibid., Eq 13); see also G. Marsh, arXiv:0711.0220v2, Eq 2 -- does not apply to 'potential reality'. The latter is needed to explain 'GR with DDE'.

In the textbook version of 'GR without DDE', the 'coordinate time' read by your wristwatch, and the very notion of 'energy conservation' during some interval from that 'coordinate time', cannot be rigorously defined. The tacit presumption is that 'space' itself does not "move" anywhere, simply because there is no "outer space" in which our 3-D space could "expand into". But only the second part from this tacit presumption is correct, since our 3-D space can indeed "move" or rather "emerge" (C. Isham, gr-qc/9310031, Option IV): the intrinsic dynamics of space -- here we enter 'GR with DDE' -- can only be defined with respect to a hypothetical global mode of spacetime, which is hidden ]between[ the "points" of the local mode of spacetime. The latter stands as a genuine perfect continuum (the "dark gaps" from the global mode are hidden by the so-called 'speed of light'), and has been studied in STR and in 'GR without DDE'.

If we wish to think as proper relativists, this is the only choice we may have: see my email to Harvey R Brown above. Many people hate it and spit insults on me; other just keep quiet, but nobody has so far offered any viable alternative.

Back in 1772, on the occasion of the fall of meteorites, the French Academy of Sciences adopted a resolution categorically rejecting such ridiculous phenomena. The obvious reason had been that rocks cannot fall from the sky, simply because there are no rocks there. Likewise, when you look at the sky, you will never think that the space itself could possibly "move", in any way whatsoever. But once the "dark energy" from 'empty space' was confirmed, the intrinsic dynamics of 3-D space, hence 'GR with DDE', are unavoidable. Your good old wristwatch does read the dynamics of "dark energy" along the arrow of spacetime.

p. 18: "The coherence between the different outcomes, in principle still accessible with the measurement of the nearly vanishing interference terms (vanishing exactly only in the limit N --> [inf]), will be totally unaccessible in practice as soon as N is sufficiently large, as the number of molecules in a bubble from the trace of a charged particle in a bubble chamber."...pp. 19-20: "The conventional picture of the measurement process in Quantum Mechanics, as an instantaneous jump from a pure state to a mixture, which affects the state all over space at a fixed time in a preferred Lorentz frame, appears, in the scenario we outlined, as the result of several limits:

"1. the time duration T of the interaction giving rise to the measurement (which, in an exact mathematical treatment, would involve the whole interval from minus infinity to plus infinity, as all scattering processes) is set equal to zero;

"2. the number of microconstituents of the amplifying part of the measurement apparatus is set equal to infinity, thus allowing exact decoherence;

"3. the volume involved by the measurement apparatus in its interaction with the system (thus occupied by the microscopic part of the apparatus) tends to the whole space, allowing the reduction of wave packets to take place everywhere;

"In the conventional picture, some form of nonlocality is unavoidable, albeit insufficient for transmission of perturbations (hence not contradicting local commutativity) or even of information [58]: for a given observer, a coherent superposition of two possibilities might be changed, instantaneously in some preferred Lorentz frame, to a state where only one possibility survives, by the measurement performed by another observer in a very far spacelike separated region....p. 21: "Thus points become fuzzy and locality looses any precise meaning. We believe it should be replaced at the Planck scale by an equally sharp and compelling principle, yet unknown, which reduces to locality at larger distances....

p. 27: "But locality is lost. There is no meaning to "E1 and E2 are spacelike separated", unless we pick a point _o_ in E, and limit ourselves to a special wedge W associated to _o_ and its spacelike complement -W. In this special case locality survives for free fields, but is bound to be destroyed by interactions on QST.

"That remnant of locality has been exploited to construct deformations of local nets for which the two particle S matrix is notrivial [70, 71], at the price of loosing locality in terms of fields localised in bounded regions. The various formulation of interaction between fields, all equivalent on ordinary Minkowski space, provide inequivalent approaches on QST; but all of them, sooner or later, meet problems with Lorentz covariance, apparently due to the nontrivial action of the Lorentz group on the centre of the algebra of Quantum Spacetime.

"On this point in our opinion a deeper understanding is needed....

p. 29: "The common feature of all approaches is that, due to the quantum nature of spacetime at the Planck scale, locality is broken, even at the level of free fields, and more dramatically by interactions. Which, as far as our present knowledge go, lead to a breakdown of Lorentz invariance as well....

"One might expect that a complete theory ought to be covariant under general coordinate transformations as well. This principle, however, is grounded on the conceptual experiment of the falling lift, which, in the classical theory, can be thought of as occupying an infinitesimal neighborhood of a point. In a quantum theory the size of a "laboratory" must be large compared with the Planck length, and this might pose limitations on general covariance. One might argue that such limitations ought to be taken care of by the quantum nature of Spacetime at the Planck scale....

"But the energy distribution in a generic quantum state will affect the Spacetime Uncertainty Relations, suggesting that the commutator between the coordinates ought to depend in turn on the metric field. This scenario could be related to the large scale thermal equilibrium of the cosmic microwave background, and to the non vanishing of the Cosmological Constant [79, 80].

p. 30: "This might well be the clue to restore Lorentz covariance in the interactions between fields on Quantum Spacetime."-------------ref. [80]: Sergio Doplicher, Quantum Field Theory on Quantum Spacetime, arXiv:hep-th/0608124v1.

p. 7: "A New Scenario. The Principle of Gravitational Stability ought to be fully used in the very derivation of Space Time Uncertainty Relations, which would then depend also on the energy-momentum density of generic background quantum states; this leads to commutation relations between Spacetime coordinates depending in principle on the metric tensor, and hence, through the gravitational coupling, on the interacting fields themselves. Thus the commutation relations between Spacetime coordinates would appear as part of the equations of motions along with Einstein and matter field Equations.

"In other words we may expect that, while Classical General Relativity taught us that Geometry is dynamics, Quantum Gravity might show that also Algebra is dynamics.

"This new scenario [12] appears extremely difficult to formalise and implement, but promises most interesting developments."---------------

If we accept the challenge of "dark energy" and "dark matter", I believe it will be wrong to use the Hamiltonian formulation of GR. It is not possible to quantize Einstein gravity in four dimensions either (Hanno Sahlmann, arXiv:1001.4188v1 [gr-qc]).

The problem is that we need to unravel a new degree of freedom pertaining to the NEW dynamics of spacetime: see R. Rakhi & K. Indulekha and S. Carroll, "more space comes into existence",

Note 1: I really cannot understand why people believe that the source of "dark energy" should necessarily have positive energy density ("Treated as a fluid, this component has negative pressure (assuming positive energy density), ... "T. Padmanabhan), and be "invariant under Lorentz transformations" (E. Santos).

In the model presented here, these two self-imposed presumptions are dropped, and the source of "dark energy" is non-existent in the local mode of spacetime. None. Zero. Zilch. You can safely set it to zero, just as you usually treat the vacuum energy density in QFT.

Yet the source of "dark energy" can take any value in the global mode, depending on the particular cosmological stage of the universe, because T (the energy momentum stress tensor of all matter and fields other than gravity, with positive energy density) can completely vanish/dissolve into the vacuum, and stay available there for any partial, full, or "over unity" recall, if needed.

All we can physically observe is that 3-D space is acting upon itself. We cannot trace back this 'energy from empty space', which is why some people called it "dark". In the same vein, the human mind would be considered "dark", because all we can observe is that the brain is being acting 'upon itself', by thinking about the brain, with the brain.

Gerard 't Hooft had a brilliant guess about the negative energy states, but his Ph.D. student Stefan Nobbenhuis ruined it, upon his guidance, by assuming that "there is no coupling other than gravitational (emphasis added - D.C.) between the normal matter fields and their ghost counterparts, otherwise the Minkowski vacuum would not be stable" (arXiv:gr-qc/0411093v3, p. 10). But the quantum vacuum can be stable without any gravitational coupling whatsoever, because gravity applies only to the explicated/physical form of reality, while the absolute value of energy stored in the vacuum is potential reality.

The misleading adjective "dark" (Mike Turner) comes from treating the potential vacuum reality as 'physically real'. In the case of John Wheeler's 'cloud', only this last explicated physical reality will be subject to gravitational interactions, like the explicatedCasimir energy. The 'cloud' has been explicated by an emerging context, while in the case of the "dark" energy from the quantum vacuum we encounter an emerging boundary: the energy here is not "boundary sensitive". It is 'the whole universe as ONE' (global mode of spacetime) that fixes the "boundary terms", so that we can only see a "cloud" with strictly positive mass. Surely the explicated "cloud" changes along the cosmological time arrow, but its "dark" driving force cannot be unraveled from the fleeting "cloud" (denoted with a in [Ref. 1]), just as we cannot unravel theUNdecidable Kochen-Specker quantum state from its colorizable projections. This is the legacy of Schrödinger and Margenau (see below), which I was hoping to explain on November 27, 2002.

Notice that in GR you inevitably need to introduce some additional structure [Ref. 2] compensating for the absence of the global mode of spacetime. To understand the origin of these of additional structures, recall the prerequisites that necessitate 'time' regarded as 'change': we need two different and distinguishable states of a physical system, and some background w.r.t.w. the difference between the two states can be identified. In STR, the task looks quite easy: take two consecutive, instantaneous, point-like states of a Frisbee along its continuous trajectory, defined w.r.t. the fixed background of Minkowski space, and you're done. In 'GR without DDE', the background is the 3-D space itself, which is allowed to change its "curvature" only, following the bi-directional "talk" between matter and space: "Space acts on matter, telling it how to move. In turn, matter reacts back on space, telling it how to curve" (John Wheeler). But because of the Equivalence Principle adopted in GR, gravity must be able to vanish completely "inside" an infinitesimal "point", hence there is no localizable point-like gravitational energy density in such "point" and across its infinitesimally small neighborhood, to compute derivatives (Bjoern Schmekel). Hence the notion of 'time', conceived as 'change' from point-like state A to point-like state B (defined w.r.t. a background of 3-D space), becomes terribly complicated because of its dual, non-linear duties: "the metric is treated as a field which not only affects, but also is (at the same time - D.C.) affected by, the other fields" (John Baez). Notice the notion suggested by Viktor Denisov and Anatol Logunov: 'physicogeometric dualism'.

Under these circumstances, people need to introduce some additional structures to define some "boundaries" of the whole spacetime; for example, some "fall-off conditions of the curvature in appropriate coordinate systems at infinity" [Ref. 2], because, in the case of the Einstein equations alone, "there are no physically motivated boundary conditions" (Alan Rendall). But notice that the background of 3-D space is still always there, in the sense that in 'GR without DDE' the space itself doesn't "move" (Robert Geroch). But once we endow the space itself with an additional degree of freedom, as in 'GR with DDE', it acquires new dynamics (arrow of spacetime), and those 'additional structures' that were introduced by hand [Ref. 2] should be derived from the "dark" ("no observable content", Domenico Giulini) global mode of spacetime.

Don't try to bridge the two "mirror" worlds (introduced by symmetries or quasi-symmetries [Ref. 2]) with any physical stuff whatsoever. Don't try to explain the self-acting faculty of 3-D space with the physical, "colorizable projections" that can only show up with positive energy density. You will be baffled by the driving force of the cosmological time arrow [Ref. 1] and the cosmological "constant" problems forever (cf. Amedeo Balbi), or at least until you quietly retire.

[Ref. 2] J. L. Jaramillo, E. Gourgoulhon, Mass and Angular Momentum in General Relativity, arXiv:1001.5429v1 [gr-qc]. To appear as proceedings in the book "Mass and Motion in General Relativity", eds. L. Blanchet, A. Spallicci and B. Whiting

p. 6: "Once the non-local nature of the gravitational energy-momentum and angular momentum is realised, the conceptual challenge is translated into the manner of determining the appropriate physical parameters associated with the gravitational field in an extended region of spacetime. An unambiguous answer has been given in the case of the total mass of an isolated system. However, the situation is much less clear in the case of extended but finite spacetime domains.

"In a broad sense, existing attempts either enforce some additional structure that restricts the study to an appropriate subset of the solution space of General Relativity, or alternatively they look for a genuinely geometric characterisation aiming at fulfilling some expected physical requirements. In this article we present an overview of some of the relevant existing attempts and illustrate the kind of additional structures they involve....pp. 9-10: "The characterisation of an isolated system in General Relativity aims at capturing the idea that spacetime becomes flat when we move sufficiently far from the system, so that spacetime approaches that of Minkowski. However, the very notion of far away becomes problematic due to the absence of an a priori background spacetime. In addition, we must consider different kinds of infinities, since we can move away from the system in space-like and also in null directions. Different strategies exist in the literature for the formalization of this asymptotic flatness idea, and not all of them are mathematically equivalent. Traditional approaches attempt to specify the adequate fall-off conditions of the curvature in appropriate coordinate systems at infinity. (...) The whole picture is inspired in the structure of the conformal compactification of Minkowski spacetime....pp. 35-36: "But it must be acknowledged (...) that the status of the quasi-local mass studies is in a kind of post-modern situation in which the devoted intensive efforts have resulted in a plethora of proposals with no obvious definitive and entirely satisfying candidate...."The moral of the whole discussion in this article is that the formulation of meaningful global or quasi-local mass and angular momentum notions in General Relativity always needs the introduction of some additional structure in the form of symmetries, quasi-symmetries or some other background structure."----------

"These gauge transformations define equivalence classes within S, which we call gauge orbits. A gauge orbit represents a unique physical state (footnote 5), and its different points correspond to different coordinatizations.--Footnote 5: "Note that this state is the whole spacetime."

"... in the space of on-shell field configurations the gauge generator moves from one point p to another p', whereas the Hamiltonian works within every point p, which already represents an entire spacetime (emphasis added - D.C.)."

In another paper (arXiv:0902.0401v1 [gr-qc], p. 4), PSS illustrated their ideas with a "spatially homogeneous isotropic cosmological model", and acknowledged that "this model possesses the curious property that the only physical variable that changes in time is time itself!" (emphasis added - D.C.).

Which makes this "time" unobservable, or rather 'observable only with respect to itself'. Pity PSS didn't ask Karel Kuchar to comment on their speculations, nor mentioned the new dynamics of 3-D space due to its "dark" energy acting on the whole spacetimeen bloc.

I think you shouldn't regret that you missed the chance to educate Kip Thorne [Ref. 1] about the inevitable failure of his LIGO project and the whole "GW astronomy". Arguments similar to yours have been spelled out by Steven Weinberg seven years ago (25 Feb 2003), after which he added: "I often find that people who say silly things actually do correct calculations, but are careless in what they say about them."

Not just Kip Thorne, but the whole LIGO Scientific Collaboration (LSC) are saying silly things, but because they use the linearized approximation of GR, they do "correct calculations".

However, if you use the same linearized approximation of GR, you can prove "GW astronomy" wrong by reductio ad absurdum:

"The gravitational deformation in general relativity does in fact belong to the kinematic effects, when it is described without the use of geodesic deviation equation. Thus, for example, the interferometric detection of gravitational waves cannot give a non-zero result, since the scales of all types of equally oriented lengths do change in gravitational fields in the same proportion, and the numbers of light wavelengths fitting along the alternative arms of interferometer cannot suffer changes in a passing gravitational wave.

"I am regretful not to tell these considerations to Kip S. Thorne more than two decades ago, simply because of a kind of awkward modesty."-----Note: See what happens if you don't tell your students everything you know.

Example: Jim Hartle. Recently, a young theoretical physicist, Gareth Jones, defended his Ph.D. Thesis on "Searching for gravitational waves... ". He looked at Ch. 23 from Jim Hartle's textbook, "Gravity, an introduction to Einstein's General Relativity", to eventually understand the dimensionless GW amplitude (Eq. 1.64, p. 15) that would "cause a periodic strain (i.e., stretching and contraction) of the proper distance between points (Sic! - D.C.) in spacetime" (ibid., p. 182).

But you can't hide the dimensionality of GW amplitude in 3-D space: check out SBG here. In order to prove LSC (at least 679 people) wrong, all you need is to drive the "background" in the linearized approximation of GR to its absurdity, as with the SBG argument. Not sure?

If you look at Wiki, GW amplitude "is not the quantity which would be analogous to what is usually called the amplitude of an electromagnetic wave (...)."

The alleged GW has frequency, wavelength, and speed -- all defined with proper dimensionality. Only the action of geometry on matter, embodied in the mantra above, is a dimensionless ghost that shows up only with ... "2.3×10-26" , say.

How can Gareth Jones change his Ph.D. brain, to think as a physicist? Surely the entity that fixes a 'meter' cannot itself be defined with what it produces -- a 'meter'. Can he notice the intrinsic parapsychology of statements like "our best (lowest) upper limit on gravitational wave amplitude is 2.3×10-26" ?

I think Jim Hartle (along with Bernie Schutz) should be blamed for Gareth Jones' professional career. It may be wasted by chasing ghosts with real, taxpayers' money.

In my opinion, the "dimensionality" of GW amplitude is just like that of quantum waves. And just like the de Broglie waves (cf. Franco Selleri above), in present-day GR these GWs cannot show up either, simply because they cannot transport energy-momentum to any physical system in the spacetime of GR textbooks.

Just a brief comment. You wrote (p. 46): "The claim of Meyer that the Kochen-Specker Theorem has been nullified leads to the question what it is exactly that the Kochen-Specker Theorem states. A common notion is that the theorem states that (at any given time) not all observables can be assigned definite values that are independent of the measuring context."

Long before John Bell, Schrödinger explained the crux of the matter (November 1950),

I suppose none of your senior colleagues have mentioned to you anything about my numerous email messages, with links to my web site. And that is not fair.

If I'm wrong about them, please write me back.

Kindest regards,

Dimi Chakalov------

Note 1: All kids, Ronnie Hermens included, have the right to know everything we know. This doesn't mean that they should accept our opinions. But in order to make educated decision about their professional career, we should kindly offer them all our knowledge and opinions. Otherwise it's just not fair.

Ronnie, here's a story from Uncle Dimi. Suppose you're in a pitch dark room. You hold a torch in your hand, but want to "see" how the darkness itself looks like before you observe it with the torch. As you might have guessed, 'the darkness itself' stands for the 'UNdecidable quantum potential reality' that cannot be fitted into any Hilbert space whatsoever; more from Uncle John.

Obviously, your task is not feasible. With your torch, you can only "see" the classical presentations of 'the dark room', which have been "filtered" through the classical spacetime at the scale of tables and chairs. So, what do you do? You use some QM textbook recipes for calculating probabilities for observing the dark room with your torch. NOT the dark room itself. And if you "look" with Hilbert space with dimensions 2, you may never notice any problems with your torch observations, as you can always infer what sorts of objects should have been in the dark room, before you turned on the torch and brought them into the classical spacetime of tables and chairs (recall the "quantum computing" mess).

The real fun with QM starts when KS Theorem comes to play: your torch observations do not make sense. You cannot infer anymore what kinds of objects should have been in the dark room before you turned on the torch. (In the context of "quantum computing", quantum states both 'partly exist' as context-evoked latent observables and 'partly do not exist', being UNdecidable quantum state (never in plural) as well. It's an "eigentümlichen, klassisch nicht beschreibbaren" (W. Pauli), indivisible bundle of 'both ONE and many'. Capiche?)

The essential ONE-"part" from the objects in the dark room will always remain unaccounted for. You can say NOTHING about it. It is UNspeakable. You just can't fit it into any Hilbert space on which you make your torch observations. It is the ultimate quantum reality 'out there', which I simply call 'potential reality'.

You can say nothing about it, because any statement of yours refers exclusively to what you can "see" with your torch. This is my answer to your question "what it is exactly that the Kochen-Specker Theorem states", as you put it.

The math jungle is staggering. Don't go there, it's endless. How do I know?Because there is only one way to solve both the measurement problem of QM and the cosmological "constant" problem. Both problems, en bloc. Just recall the basic tenet of Copenhagen [Ref. 1], and compare it with the interpretation of QM discussed here and the arrow of spacetime: the instantaneous state of Die Bahn (trajectory) of all physical objects, at all length scales, is nothing but the local mode of spacetime, as it evolves from "point" A to "point" B, following the expansion of space due to the "dark" energy of the vacuum.

Locally, Die Bahn [Ref. 1] is like a single flash or "point" with infinitesimal duration. The whole 'local mode of spacetime' is comprised of such already-correlated "points" with infinitesimal duration, all of which are explicated with a carpe diemunit probability, in line with the PR interpretation of QM. If you can imagine an instantaneous cross-section of the arrow of spacetime, the local mode will be a frozen flat 3-D space with total gravitational energy precisely zero (see above): the potential reality itself (the infinite "shop", see below) does not gravitate (cf. "yes you can"). Any of these 3-D "snapshots" constitutes a legitimate 'universe' by itself, but as the arrow of spacetime "moves" forward, we are led to consider a chain of such snapshots, and then it looks like the universe is somehow 'acting upon itself', because the "dark gaps" from the global mode are precisely sealed off, making the local mode a perfect continuum (notice the emergence of "waves" below). Of course we cannot locate the source of this "dark energy", call it X , inside the very thing that is being produced by X : the 3-D space of the local mode of spacetime. Which is why some (otherwise serious) people called it "dark", while other speculate about some mysterious "dark flow" and "axis of evil", as pictured here.

A few days ago, I tried to explain Die Bahn [Ref. 1] to my teenage daughter (she is the 'test bed' for my
DVD video tutorials) as follows. Picture the quantum vacuum as an infinitely "large" cash-and-carry shop, from which you can get everything and anything (that caught her attention, as expected). You buy some stuff for your diet, but you can shop (i) only if you're "off the train", and (ii) only for your one-day meal. This 'one-day meal' is just one point from a perfectly legitimate universe -- the local mode of spacetime at the global instant 'now'.

Such "horizontal" cross-section of the arrow of spacetime takes only one sliding point 'now' from the ("vertical") arrow of spacetime. But because you inevitablychain your 'daily points' along a perfectly continuous trajectory (the "dark gaps" of the global mode of spacetime are completely sealed off due to 'the speed of light'), you get an emergent Bahn with 'potential future' and 'irreversible past', as it emerges due to the arrow of spacetime, while the infinite shop, with all its (latent) stuff that you did not buy, will always remain "dark".

Imagine also that every day you're buying more stuff: if some theoretical physicists compare only your daily meals during a week (they can't glance at the "dark gaps"), they will claim that you've been getting 'more stuff' throughout the whole week from some "dark source". (If they include the infinite shop in their calculations, they will inevitably face their cosmological "constant" paradox.)

Also, notice that all people in the universe are shopping from the same "dark shop": they shop for their 'one-day meal' to cook up their common 'local mode of spacetime', but have to do it relationally (there is no background whatsoever in the local mode of spacetime), and also in a pre-correlated (EPR-like) fashion, in line with the Bootstrap Rule 'think globally, act locally'. The result is a (quantum-gravitational) wave of pre-correlated 'one-day meals' that also emergesduring Die Bahn. Nothing is "waving" to emit these waves. They just emerge. Capiche?Okay, let me try it this way. Do you remember how we played Frisbee on the beach? That poor Frisbee was too large and heavy to take advantage of its potential UNdecidable KS states, and since it also managed to ignore the effects of gravity, we could imagine some fictitious "fixed grid" (Minkowski spacetime) on which we can draw its Bahn. Had the Frisbee been a quantum-gravitational object, it would have all its instantaneous one-point "meals" correlated (recall the two hands) with 'everything else in the universe' ("off the train", in the global mode of 'the Frisbee per se'). Then the real Frisbee would be flying like a fish from a correlated shoal of fish, only you can't see the rest of fish from the shoal, and therefore can't see its wave-like movement, like the waves of the correlated centipede's legs. Simple, no?

I didn't try to explain why LSC may only unravel the blueprints from relic GWs cast in the distant past, but cannot observe 'the emergent wave' online, as it evolves along the arrow of spacetime. That would have been too much for her.

Finally, notice that if you take the whole stack of "Photoshop layers" (all your 'one-day meals') and flatten them onto one (.JPG) image, you will end up with a timeless "trajectory" immersed in a dead frozen 4-D "block universe", in which "time" can evolve just as much as "space" can: "There is no dynamics within space-time itself: nothing ever moves therein; nothing happens; nothing changes" (R. Geroch). Our (mischievous) wristwatches do in fact read the global cosmological time of 'Die Bahn', but because you can only use a physical "torch", you are deceived by a flatten image of all 'one-day meals': in Fig. 1 below, there are three consecutive 'one-day meals' or "points" from Frisbee's trajectory, connected by the pre-geometric plenum, while Fig. 2 shows the stack of "Photoshop layers" consisting of three different, re-created 4-D universes in their local mode, as they evolve along Die Bahn (the red arrow of spacetime, w ). The quantum-and-gravitational waves "travel" along w (Fig. 2), which is why they cannot be detected on the flatten image (Fig. 1): regarding GWs, the linearized approximation of GR is a "shadow without power" (Hermann Weyl).

Fig. 2Notice the three red 'one-day meals', or three 'sayings', beingconnected by the pre-geometric plenum; its quantum versionis the UNdesidable KS state. In GR, the pre-geometric plenum"connects" all points from the local mode of spacetime with'the undivided universe as ONE', which is being multiplied asinfinitely many points in the local mode.

With the current GR and diff geometry textbooks, you can explore only one instant 'now' from the arrow of spacetime. Even if you try to introduce some "infinitesimal coordinates shift" [Ref. 2, Eq. 4.1], as in today's GR textbooks, you will nevertheless be dealing with a frozen one-instant "trajectory", and can never solve the Cauchy problem for the Einstein equations: the "dark background" of 'the reference fluid' of GR is missing from GR textbooks. You can use such 'calculated-from-one-instant trajectories' only in classical physics and STR (e.g., the trajectory of a Frisbee, calculated from some of its instantaneous states).

To resolve the genuine dynamics of GR, we need to make the 3-D space dynamical: the global cosmological time originates from the dynamics of space. First of all, we have to replace the familiar expression 'with respect to the rest of the Universe' [ibid., p. 263] with a Machian "absolute" reference frame, which works as 'the reference fluid' of GR -- the global mode of spacetime. Recall also the idea of "breathing" (inhaling/exhaling) Universe, and imagine the elementary step/cycle of the arrow of spacetime (cf. Fig 2 above) as the "negotiation" of the two hands in Escher's drawing below: an inhaling ("offer") quantum-gravitational wave is being emitted in the global mode of spacetime from each and every "point" from the local mode, followed by an exhaling ("confirmation") quantum-gravitational wave converging on the next point from the next horizontal layer.

What is the duration of this "breathing" cycle in the local mode of spacetime? ZERO. This is the meaning of the phrase 'Your Global Time is ZERO'. The local mode of spacetime is a perfect continuum. But that's too much for you, isn't it?

p. 263: "The principle of space travel while locally “at rest”, is analogous to galaxies receding away from each other at extreme velocities due to the expansion (and contraction) of the Universe.

"Instead of moving a spaceship from a planet A to a planet B, we modify the space between them. The spaceship can be carried along by a local spacetime “singular region” and is thus “surfing” through space with a given velocity with respect to the rest of the Universe.........§1.2.1, 'The (3+1) Formalism: the Arnowitt-Deser-Misner (ADM) technique'

p. 265: "In 1960, Arnowitt, Deser, and Misner [5] suggested a technique based on decomposing the space-time into a family of space-like hypersurfaces and parametrized by the value of an arbitrarily chosen time coordinate x4.

"This “foilation” displays a proper-time element dτ between two nearby (emphasis added - D.C.) hypersurfaces labelled x4 =const and x4 + dx4 = const. The proper-time element dτ must be proportional to dx4.........p. 267: "The main advantage of the ADM formalism is that the time derivative is isolated (emphasis added - D.C.) and it can be used in further specific computations.........p. 275: "This horizon first appears for the occupants of the spaceship, who are unable to “see” beyond the distortion, and therefore cannot communicate with the outer universe.........p. 280: "Let us consider the infinitesimal coordinates shift

I greatly admire your Extended formulation of GR (EGR), particularly the so-called “residual” (true) field tensor. Yet it seems to me that the infamous "infinitesimal coordinates shift" in ADM presentation, which you used in your latest article cited below, is seriously flawed, as hinted in the subject line.

You said (Feb 16th) that "will take a look" at my note on KS Theorem, yet in your latest arXiv:1001.2867v3 [quant-ph], co-authored with John Cramer, there is not even a hint to it.

Regarding the UNabsorbed offer wave, you and John argued (footnote 9) that "it is the uncommitted (to a particular basis) nature of the offer wave which gives it its flexibility and thus its ability to explore “all possibilities at once.” "

How would you (and John) tackle these 'all possibilities at once' if they pertain to the UNdecidable KS state? How can you, or anyone else, derive the Born Rule in the case of KS state?

Can you possibly derive *anything* resembling 'probabilities' in the case of KS state?

The "infinite" velocity of the "phase wave aspect" corresponds to the *global mode of time* in which the "talk" b/w matter and space "take place": "Space acts on matter, telling it how to move. In turn, matter reacts back on space, telling it how to curve" (John Wheeler),

Note: Nothing in the local mode of spacetime can (nor have to) wait "... until the conserved quantities are transferred and the potential quantum event becomes real", as John Cramer put it.

We see only the end result from already-completedatemporal "handshaking" transactions, and of course we cannot trace it "back" to the global mode of spacetime -- all the negotiations between the offer-and-confirmation wave is one single event in the local mode of spacetime. In the local mode of spacetime, its "duration" is zero -- just like the proper time of photon's flight "between" its emission-and-absorption; see Kevin Brown.

The second case of fixing the physical constituents at this same "point" concerns GR: we need to examine the "negotiations" of matter and geometry, which also "take place" in the global mode of spacetime: "Space acts on matter, telling it how to move. In turn, matter reacts back on space, telling it how to curve" (John Wheeler).

In GR, the offer-and-confirmation wave pertains to the "dipole radiation", and as it "converges" on the next point from Die Bahn above, it fixes the positivity of mass in the local mode of spacetime. According to
RSP, we can imagine "two" such waves only in the global mode, as they "run" in two opposite "directions: from the macro-world of classical mechanics toward S and L , and vice versa.

In my (perhaps very biased) opinion, these "two" (offer and confirmation) waves should determine the instantaneous inertial reaction "forces" as well. Currently, GR says nothing about the origin and mechanism of (Machian) inertial "forces".

Regarding the geodesic hypothesis, Alan Rendall acknowledged: "In elementary textbooks on general relativity we read that the Einstein equations imply that small bodies move on geodesics of the spacetime metric. It is very hard to make this into a mathematically precise statement which refers to actual solutions of the Einstein equations (and not just to some formal approximations)." Notice that Alan Rendall didn't even mention those 96% of the stuff in the universe, which is "dark" and moves on some weirdly modified geodesics.

Yes, Sidney Harris is right: the re-creation of the local mode of spacetime, along the arrow of spacetime, does look like a "miracle", perhaps because we know nothing about the so-called 'speed of light' that is hiding the global mode from any direct view from the local mode. All we can observe is 'the world of facts' with unit probability, which has already, post-factum being cast in the past, as in the example with the Sun: we cannot observe the actual state of the Sun, but only its state that has been 'actual' some 8 min before we looked at the Sun.

The local mode of spacetime is a perfect continuum, because the "dark gaps" from the global mode are completely sealed off by the so-called 'speed of light'. We also have a pocket of propensity-states (KS states; see below) to choose from, which cannot be fully derived from our past only -- this is the lesson from the Free Will Theorem. Sorry for repeating this all over again.

Note: Let me quote from Robert Wald's arXiv:gr-qc/0511073, "Teaching General Relativity", p. 5 (emphasis added): "First, one needs a mathematically precise notion of the "set of points" that constitute spacetime (or that constitute a surface in ordinary geometry). The appropriate notion is that of a manifold, which is a set that (...) but has no metrical or other structure. The points of an n-dimensional manifold can thereby be labeled locally by coordinates ... ."

This last sentence is a total Jabberwocky. How did we get these "points" that can be "labeled locally by coordinates", such that we can, and also have to, shuffle these coordinates to extract some GR observables?

Contemporary relativists begin by postulating a Hausdorff topological space that has been somehow "connected", but usually mention this miracle in footnotes (e.g., Chris Isham, Modern Differential Geometry for Physicists, p. 61, footnote 1). They would denote this connected topological space with M , and introduce a second postulate: a differential structure on M , such that M is promoted to a four-dimensional manifold. Then the third step is obvious: introduce a metric, and they are ready to teach GR (cf. Diego Meschini et al., Sec. 2.1) and muse over the exotic differentiable structures and the puzzle posed by Carl Brans:

Perhaps one first needs a mathematically precise notion of the "set of points" that constitute a line (1-D Euclidean space), to elucidate how these "points" are connected in such way that the law of continuity, as defined in the standard calculus texts of the 1800's, is fully obeyed: the consecutive points of the same line should succeed each other without any interval or 'point in-between' them.

However, how should the topological space itself been "connected", in order to evolve into a perfectly smooth manifold? Specifically, does there exist any 'smoothness structure' on any topological manifold? I'm not aware of such beast in differential topology textbooks; hence my email above.

But here's the catch: on the one hand, the 'point-connecting agent' (called here pre-geometric plenum), which makes aHausdorff topological space "connected", must not be anything that may, in any imaginable way, exist within the line made by "zero-dimensional" points. It can't be some "special middle point" or "special separating interval", because inside a line we have only points, and nothing but points. These points don't have any special hooks or handles that can determine the principle of locality and the so-called "speed" of light.

On the other hand, the pre-geometric plenum must somehow exist in order to "separate" the points and preserve their individual, albeit fleeting, physical content: "For example, \phi is a scalar field on the manifold and $X$ represents the space-time coordinate of a particle, then although \phi(x) has no physical meaning (if x is a point in the space-time manifold) nevertheless $\phi(X)$ *does* have a meaning: ie you can talk in a Diff(M)-invariant way about the value of a field where a particle 'is', and similarly for a trajectory" (Chris Isham, private communication).

The only way out from this conundrum is to use the new (to contemporary relativists) form of reality, as explained in the case of the human brain and the quantum world. The potential reality does not exist in the local mode (cf. Fig. 1), hence the latter is a perfect continuum -- the "gaps" are completely sealed off by the so-called speed of light, because the "duration" of the transition from one "point" to the nearest "point" is zero: check out Kevin Brown above.

All this may sound like some metaphysical exercise devoid of any mathematical implications, but recall that in mathematics you have to follow the obvious and intuitively clear metaphysical ideas, or else will sink in a jungle with no way out.

Not to mention the infinite amount of energy packed in the quantum vacuum.

But how would you “quantize” a Riemannian manifold [Ref. 1, p. 425] and gravitational "field" represented by Riemannian metric? How would you specify a Poisson structure of the "dynamical system", as driven by [we-do-not-know-it]?

Please notice that the UNdecidable KS state (called 'potential reality') has been interpreted as 'pre-quantum reality'.

For comparison, a similar task has been undertaken by Antony Valentini, at the expense of introducing some "signal nonlocality" [Ref. 1] and "hidden variables" [Ref. 2].

Unlike Valentini's theory, I trust every physicist can check out the interpretation of KS Theorem at the link above, and find out whether there is an error in it.

Should you or any of your colleagues find an error, please do write me back. If you cannot find such error, please be assured that I would be delighted to attend your Workshop and explain the so-called 'PR interpretation of QM'.

"Valentini has been working on an extension of David Bohm's "ontological interpretation" of quantum theory that would allow "signal nonlocality" that is forbidden in orthodox quantum theory. "Signal nonlocality" allows nonlocal quantum entanglement to be used as a stand-alone communication channel without the need of a classical light-speed limited retarded signal to unlock the entangled message from the sender to the receiver. This would be a major revolution in physics ... "

"What is so unusual about Antony Valentini? Just this: he's resurrected a theory that undoes the central tenet of quantum mechanics, and gives relativity theory a good punt to left field as well. The theory follows quantum math, but at the same time allows for new possibilities beyond conventional quantum mechanics.. It's a theory that says there is indeed an objective reality behind the things we observe -- that quantum uncertainty is not fundamental. And that somewhere, somehow, time is universal -- not relative. Goodbye, ghostly probabilities, with their strange propensity for collapsing into real things while apparently sort of holding back and remaining always a bit coy and ghostly... hello, hidden variables that are objective.

"And Antony's particular twist on the theory suggests a new explanation for the uniformity of the early universe -- where, he suggests, quantum law might not have applied, where stuff could interact faster than the speed of light -- and where those interactions were actually visible.

"There's no proof, of course -- at least not yet. But it's fun to think about.......

"Q: How are you going to convince anybody of this?

"A: We need to find a violation of quantum mechanics in the early universe. We need to find a non-quantum distribution of particles. There may be particles floating around in space now which were left over from that very early time. People are looking for particles of dark matter left over from the early universe, and some of these may be good candidates. Another possibility is relic gravitons, particles associated with gravity that are believed to have stopped interactingwith other particles at a very early time. Perhaps these relic gravitons from the early universe don't obey quantum mechanics.........

"Q: How does the pilot wave theory view time and space?

"A: It goes against relativity theory, because it has faster-than-light processes, and in relativity nothing is supposed to go faster than light. So it seems to me that we may have to revise relativity theory and end up with a notion of universal time. In relativity, different observers at different speeds have their own time and there is no absolute time. But in this theory, distant observers can communicate instantaneously if they have control at this fundamental level of non-quantum particles. So they would be able to synchronize their clocks instantaneously even if they were millions of light years apart. Of course, some people don't like the idea, and that's a problem."---------

Note: I haven't yet received feedback from Mike Towler and from his colleagues regarding my email sent three days ago. Surely the KS Theorem poses very tough challenges regarding the crux of 'quantum reality'.

"... the implication of the discussion above is that the value ascribed to B (resp. the result of measuring B) depends on whether it is considered together with A1, or together with A2. In other words the value of the physical quantity B is contextual. This is often considered one of the most important implications of the Kochen-Specker theorem."

The immediate question is this: What remains invariant in 'the quantity B' upon changing its "color" (see 'KS Theorem for teenage girls' below), to qualify as 'quantum reality of the quantity B'?

My proposal: pre-quantum reality. In the framework of Chris Isham's approach, the so-called pseudo-states (ibid., p. 16) are ‘as close as we can get’ to the UNspeakable pre-quantum reality (called here 'potential reality'). In the case of the human brain, you can "measure" the latter with three (or more) sayings ("pseudo-states"), yet can never "collapse" the ultimate potential reality from which these "contextual" quantum states emerge.

Regardless of how you tackle such "pseudo-states" with some topos approach, the solution to the measurement problem in QM requires that you offer some 'peaceful coexistence' (Abner Shimony) of this pre-quantum reality and STR, as explained at this web site. Recall its motto: Dead matter makes quantum jumps; the living-and-quantum matter is smarter. We could have sorted out this bundle of issues eight years ago, but I guess Chris Ishamhad a different agenda.Anyway; here's an anecdotal story from 1970s (ibid., footnote 3, p. 6):

"I have a fond memory of being in the audience for a seminar by John Wheeler at a conference on quantum gravity in the early 1970s. John was getting well into the swing of his usual enthusiastic lecturing style and made some forceful remark about the importance of the quantum principle. At that point a hand was raised at the back of the lecture room, and a frail voice asked “What is the quantum principle?”. John Wheeler paused, looked thoughtfully at his interlocutor, who was Paul Dirac, and answered “Well, to be honest, I don’t know”. He paused again, and then said “Do you?”. “No” replied Dirac."

If you, my dear reader, cannot find an error in the interpretation of KS Theorem below, I will be happy to offer you my version of 'the quantum principle' and the origin of the quantum of action.

> As I'm sure you are aware, sending unsolicited emails to large numbers of> famous people asking what they think of your theory is not only widely> considered to be appallingly rude but to be the hallmark of a crackpot.

It is not about my "theory", as you put it.

I'm afraid there is a large number of "famous people" who ignore the legacy of Schrödinger and Margenau, and cannot grasp the basic basics of KS Theorem and CK Free Will Theorem,

If you and the rest of these "famous people" consider this "appallingly rude" and "the hallmark of a crackpot", I can only wish you a quiet a peaceful retirement.

Alternatively, if you and the rest of these "famous people" wish to get professional, please don't hesitate to write me back, with *specific arguments* refuting the interpretation of KS Theorem offered at the link above.

On Thu, 25 Nov 2010 00:03:37 +0200, Dimi Chakalov <dchakalov@gmail.com> wrote:>> Hi Carlo,>> I quoted your statement in the subject line at>> http://www.god-does-not-play-dice.net/Margenau.html>> If you disagree with my critical remarks and/or wish to comment on my> proposal, please don't hesitate to write me back.>> I will be happy to hear from your colleagues as well. Will keep the> discussion private and confidential -- provided it is professional.>> All the best,>> Dimi>

"The day may come when quantum networks are ubiquitous. An advantage that steganography has over standard encryption schemes is that private information could be transmitted over a long time through a network, completely undetected and even unsuspected by other users. Steganographic techniques may also be useful as a way of authenticating quantum communications in distributed quantum information processing; such uses of classical steganography for authentication are often called “watermarking.” Quantum steganography has far reaching consequences, and may provide a measure of security beyond what classical steganography can afford.....

"Alice and Bob conceal their communication from Eve, hiding their message as errors in a codeword for an “innocent” cover state |psi_c>, and using the resource of a shared secret random key. (Shared entanglement would work as well, or even better.)"

Note: Recall also the elusive Event Horizon -- a global property of an entire spacetime, which should be somehow "defined nonlocally in time" (J. Thornburg, lrr-2007-3). The mythical "black holes" cannot be defined rigorously in GR, so try the global, Heraclitean, and non-Archimedean time of the UNdecidable quantum state of 'the universe as a brain'. Or trust Chris Isham.

p. 2: "Weyl’s principle: The world lines of galaxies, or ‘fundamental particles’, form (on average) a spacetime-filling family of non-intersecting geodesics converging towards the past.

"The importance of Weyl’s principle is that it provides a reference frame based on an expanding ‘substratum’ of ‘fundamental particles’. In particular, if the geodesic world lines are required to be orthogonal to a series of space-like hypersurfaces, a comoving reference frame is defined in which constant spatial coordinates are“carried by” the fundamental particles. The time coordinate is a cosmic time which labels the series of hypersurfaces, and which may be taken as the proper time along any of the particle world lines."----------

It is the arrow of space (AOS) that makes 'more and more space' to emerge (Isham and Butterfield) from [we-do-not-know-it], hence "the distances between all elements of the cosmic substratum (or, fluid) grow with time" (Michal Chodorowski), and we enjoy 'arrow of spacetime'. Were it possible to physically trace back the entity called [we-do-not-know-it], it won't be "dark" anymore, and the Aristotelian First Cause would be shifted one step further.

Notice that the arrow of space (AOS) leads directly to Machian quantum gravity, as the motion of any individual body is to be defined with respect to the entire universe (E. Mach, The Science of Mechanics (1883), Open Court, 1960, pp. 286-287). It has "infinite extent" (J. Barbour, arXiv:1007.3368v1 [gr-qc], p. 26), being in the state of ONE-ness (global mode of spacetime, cf. Fig. 2) that keeps "the last remnant of physical objectivity" (A. Einstein).

Physicists hate the Aristotelian metaphysics, however. They relentlessly try to picture [we-do-not-know-it] as some physical stuff with positive energy density, and end up with searching for an 'elephant in a china shop', only to find out that the elephant must be many orders of magnitudes larger than the store itself.

The AOS-driven dynamics of living and quantum/gravitational systems will inevitably produce a self-acting action, because the non-linear bi-directional negotiation between every "fish" and 'the rest of fish from the shoal' is "dark" in the local mode of spacetime. Sorry for repeating this again; I know it's boring.

If you disagree with the arrow of space (AOS), just try to define quasi-local quantities in asymptotically flat spacetime wrapped with (flexible?) "boundaries" at spatial infinity, yet keeping the splitting of spacetime (ADM) into two entities, one of which (called 'time') would refer to things that "evolve" with respect to something fixed, called 'space'. As R. Penrose acknowledged in November 1981 [Ref. 2], "several problems of interpretation remain to be solved".

Fuggedaboudit, Roger. Time can "evolve" just as much as space can; hence the arrow of space viz. arrow of spacetime endowing the
conservation of quasi-local observables of Type I matter fields (Eq. 1 below), bootstrapped by gravity. Direct observation of "pure gravitational field" (cf. Dupre and Tipler below) is like direct observation of the human mind, while acting on its brain. Gravity makes all matter fields self-interacting, hence the proper GW detector should be endowed with the faculty of self-acting, that is, capable of acting on its own potential states along the arrow of spacetime.

According to today's GR (Mario Goto et al., arXiv:1007.4846v1 [gr-qc]), "the Strong Equivalence Principle postulates that at every space-time point in a arbitrary gravitational field it is possible to choose a locally inertial coordinate system such that, within a sufficiently small (notice the poetry - D.C.) region of the point in question, the laws of the nature take the same form as in unaccelerated Cartesian coordinate systems in the absence of gravitation. On the other hand, the Weak Equivalence Principle is nothing but a restatement of the observed equality of gravitational and inertial mass."NB: When and how does 'the finite small' shift to "sufficiently small", such that (operational definition) you "may erect a locally inertial coordinate system in which matter satisfies the laws of special relativity" (Steven Weinberg, pp. 62-68)? The laws of STR are applicable only if the so-called "sufficiently small" has already become a bona fide geometrical point from the global, Heraclitean, and non-Archimedean realm. On the other hand, the effects of gravity apply only to the Archimedean realm of finite things, such as 'one second' (see the drawing below) or 'one meter'. The bi-directional "talk" of matter and geometry ("space acts on matter, telling it how to move; in turn, matter reacts back on space, telling it how to curve", John Wheeler) is the ultimate "talk" of the Archimedean (local) and non-Archimedean (global) realms of spacetime. The cornerstone puzzle of GR is that your wristwatch does indeed read this "talk", and the covariant divergence of the stress-energy tensor does indeed disappear, or rather "dissolves" in the purely geometrical, non-Archimedean realm of "sufficiently small", staying available to re-emerge, as a quantum-gravitational phenomenon.

If you disagree with the pre-geometric plenum, try to 'connect the dots' in the drawing of 'one second' by using only Archimedean geometry and physical stuff that is invariant under "active" diffeomorphisms. Or explain the vanishing of the covariant divergence of the stress-energy tensor. Good luck.

p. 259: "By definition, Tuv(Matter) describes all the local energy, so any energy due to the [Weyl tensor] must be nonlocal in character. The existence of nonlocal energy is also manifest in the fact that the local
conservation law

Tuv^v = 0

is a covariant curved space equation. One cannot just integrate Tuv(Matter) over a 3-space to obtain a conserved total energy-momentum. The total energy-momentum of a system must, therefore, involve
nonlocal contributions due, perhaps, to the presence of Weyl tensor or to nonlocal interactions (e.g. Newton's potential energy) between the Tuv's at
different points (...). The nonlocality of the gravity energy was then exhibited in the local dependence of the pseudo-tensor on the choice of coordinate system."

p. 458: "The contributions of gravity to
energy-momentum conservation should somehow enter non-locally as corrections to the calculation of total energy-momentum. (...) From this perspective, gravitational contributions to energy-momentum, in a sense, ‘slip in through the
cracks’ that separate the local equation [XXX] = 0 from an integral conservation law of total energy momentum.....p. 777: "Thus, any non-constancy in [lambda] would have to be accompanied by a
compensating non-conservation of the mass-energy of the matter."

"Almost a century after the birth of GR there is yet no universal consensus on how energy, momentum and other conserved quantities should be defined in it from a fundamental viewpoint. (...) The main reason to defend covariant conserved quantities in GR is that, according to the general covariance principle, if conserved quantities were intrinsically non-covariant they would be irrelevant to the description of Nature....."To be precise, the general covariance principle claims that the description of Physics can be done independently of any a priori coordinate fixing. It does not exclude that in particular situations one has a posteriori preferred coordinates, preferred splittings between space and time, or preferred observers; see [2], [3], [4], [5]. One very well-known example of such a situation is Cosmology: in Friedmann-Robertson-Walker solutions one has canonical clocks (e.g. the temperature of the cosmic background radiation) that not only break Lorentz invariance defining a cosmic (global) time but break the Galilei invariance defining observers which are at rest with respect to the cosmic background radiation."

"Via their energy-momentum density, material sources generate gravitational fields. Sources interact with the gravitational field locally, hence they should be able exchange energy-momentum with the gravitational field -- locally. From this physical conception we are led to expect the existence of a local density for gravitational energy-momentum."

"Most of the leading relativists in the early twentieth century, for examples Eddington [18] and even Einstein himself [19], claimed that general relativity was an ęther theory, but they gave no mathematical demonstration of their claim.....

"According to Einstein, in his Autobiography [12], the most natural choice for the tensor Sμv is the stress-energy tensor. Einstein was uncomfortable with adding the term [xxx] to the Ricci tensor, saying it was only introduced for 'technical reasons,' required by the vanishing of the covariant divergence of the stress-energy tensor.........

"The vanishing of the divergence of the stress energy tensor is derived in Minkowski space using all the symmetries of Minkowski space. But leaving Minkowski space for a general spacetime means losing the symmetries that allowed the derivation of T [xxx] = 0 to start with!......

"As MTW emphasize, the requirement that there is no “prior geometry”— that the metric is entirely determined by the field equations for gravity — actually fathered general relativity......

"A central point of Lorentz’s 1904 paper, in which he derived the Lorentz transformations, was that the Maxwell equations — for Lorentz, the equations of the ęther — do not allow an absolute time to be defined. This is of course now obvious since the speed of light in the vacuum is a constant, independent of a inertial observer.

"So the ęther can be thought of as defining a time direction different from what we may have thought of as Newtonian absolute time........

"If space is not spatially flat, then the spatial Riemannian metric will define a metric connection, and we might thus have two connections, one from the spatial metric, and one in the time direction only......

"We suspect, but do not attempt to prove, that maintaining the distinction between two such connections would be very difficult.

"Essentially, the requirement that the connection arise entirely from the metric is nothing but the “no prior geometry” assumption, which, as we pointed out earlier, is the only assumption that will allow the geometry to bedetermined by the matter distribution and the boundary conditions. Once again, MTW have emphasized that the“no prior geometry” assumption is the basic assumption of general relativity. It is also an essential assumptionof the curved ętherial Newtonian gravity theory we develop here.......

"The question is, what should we select for the tensor Sμv . According to Einstein in his Autobiography: “Onthe right side [of the Einstein equations] we shall then have to place a tensor also in place of [the mass density] . Since we know from the special theory of relativity that the (inertial) mass equals energy, we shallhave to put on the right side the tensor of energy-density— more precisely the entire energy-density, insofar as it does not belong to the pure gravitational field ([12], p. 75.)."

It had been suggested to Einstein by Levi-Civita, who had pointed out that, by virtue of Bianchi identities, the covariant divergence of the stress-energy tensor of matter and fields *has to* be equal to zero, in order to satisfy the dynamical laws of continuous media, as known in 1915:

Hence we enjoy 'arrow of space', driven by some "dark" [we-do-not-know-it]. The (covariant divergence of the stress-energy tensor of) matter and fields can *completely* vanish/dissolve into the quantum vacuum, and stay available there for any partial, full, or "over unity" recall, if and when needed.

All this requires a new form of reality, after Schrödinger, Margenau, and KS Theorem:

Note: Since we represent matter by "a wooden nose in a snowman" (A. Einstein), what actually "vanishes" is completely outside present-day GR. Perhaps it is safe to say that the confusion about what becomes 'quasi-local' due to gravity, and exactly how, is enormous -- check out Carl Hoefer,Roger Penrose, Babak and Grishchuk [Ref. 1], and Einstein's Equivalence Principle (Okon and Callender; Hans Ohanian): the wegtransformierbar faculty of gravity (Hermann Weyl) over a "point", in the non-Archimedean realm of 'the grin of the cat without the cat', as observed by Alice.

Both the Riemannian space and Minkowski space can only accommodate facts. In the latter case, you have insurmountable problems with reconciling QM with STR. In the former case of Einstein's GR, the only event that can qualify as 'fact' is the already-completed bi-directional "talk" of matter and geometry over a "point". This is the origin of "the laws of an instant" (Karel Kuchar). But we may derive dynamical laws from 'an instant' only in Minkowski space; see what happens in GR here and here.

To be precise: I believe there exists a concealed, yet-to-be-identified object, which plays a dual role in GR, as it shows up as either "components of the metric tensor" or "gravitational field variables" [Ref. 1]. In plain words: "the metric is treated as a field which not only affects, but also is (at the very same instant - D.C.) affected by, the other fields" (John Baez). And from Laszlo Szabados: "the metric has a double role: it is a field variable and defines the geometry at the same time" . Therefore, if you employ some classical space that can only accommodate facts, and try to apply the Equivalence Principle, you are destined to a blind alley: on the one hand, the "ether" must not "come back" (M. Montesinos), but on the other hand -- the gravitational (field) energy "contributes non-locally to the total energy" (R. Penrose), and you're back in murky waters, since November 1915. People are very reluctant to acknowledge that the gravitational "field" is not a classical field. Instead, they either keep quiet (Chris Isham) or offer their "pearls" of wisdom, like Gerard ’t Hooft.

I think the introduction of some "flat space" in GR (CEOFOP, p. 25), as well as a "true, real stress-energy-momentum tensor for gravity" (see above), is not even wrong. Yet such ideas deserve publishing, because students should be made aware how vulnerable the mathematical formulation of GR is to ridiculous ideas -- see CEOFOP's "pearls" above.

Let me try to explain my viewpoint, in the framework of 'the universe modeled as a brain'. I take for granted that matter can interact with matter only. Corollary: any direct action of geometry on matter (e.g., Feynman's “sticky beads”) is banned. It is like direct action of the human mind on its brain or other physical systems. In this context, the action of the alleged GW strain on physical bodies (LIGO's arms) should be considered 'GW psychokinesis'. To avoid such parapsychology, we should investigate how matter interacts with matter in a Machian universe, in which the non-linear negotiation and feedback from 'everything else' is encoded in the emergenceof what has been called in GR 'geodesics'.Regarding GWs: the omnipresent "direction" of GW propagation takes place in the global mode of spacetime; it correlates every "fish" with the whole school of fish, hence such AOS-driven dynamics will produce an emergent quasi-local geodesic of every "fish", and will induce geodesic waves, much like the waves of the legs of a centipede. Of course, we are confined in the local mode of spacetime, and cannot observe these emergent geodesics waves.

The "Gespensterfelder" (EPR-like) "action" from 'the whole school' on every quasi-local fish will show up as "dark", because it cannot be traced back from any quasi-local fish. LIGO is not endowed with the faculty of 'self-acting', and cannot detect such "dark energy from empty space".

Recall the game of '20 questions', courtesy from John Wheeler [Ref. 2]. The quasi-local object 'cloud' cannot be represented by a tensor, because it is not an 'objective reality out there'. It brings the quasi-local quantum-gravitational contributions -- just the contributions -- to matter and fields in the r.h.s. of Einstein equation. These contributions are being converted, in the global mode of spacetime, to bona fide type I matter fields; they just acquire an additional degree of freedom due to the bootstrapping faculty of gravity, resembling the geodesic hypothesis (A. Rendall) in today's GR (summary from K. Koehler,B. Mashhoon, and N. Dadhich).

To explain these quasi-local quantum-gravitational contributions, think of the object 'cloud' as a fish from the school of fish: at each and every instant from their collective quasi-local "geodesics", we have
local conservation of energy and momentum to every closed (finite infinity, G.F.R. Ellis) system [matter & cloud], but this local conservation pertains only to one "horizontal" snapshot from the arrow of space -- cf. Figs 1 and 2 from 'Die Bahn'.

Thus, "the covariant divergence of the stress-energy tensor" (Wiki) does indeed vanish, because at each and every instant from the arrow of space (hence arrow of spacetime) the bi-directional talk of matter and geometry (cf. the double role of Einstein's equations, M. Montesinos) has been already completed, and in such already-correlated instantaneous snapshot all "nonlocal" and "dark" stuff has totally vanished.

Hence the "freely falling" bodies can indeed follow geodesics, as "the stress-energy has zero divergence" (Kenneth R. Koehler) at each and every instant 'now' from the local mode of spacetime (cf. Fig. 1). Picture these "horizontal" (local mode) sections of the arrow of spacetime as Photoshop layers stacked on the w arrow (cf. Fig. 2 above): the requirement 'stress-energy must have zero divergence' is indeed fulfilled, along with the Equivalence Principle, but only "during" an instant 'now' from the local mode.

However, because we inevitably flatten all "Photoshop layers" due to the so-called "speed of light", we see a perfect continuum of already-correlated facts, chained along a perfectly continual trajectory (e.g., the trajectory of a Frisbee on Minkowski space) or perfectly continual geodesic. In order to follow a geodesic(Alan Rendall), the Frisbee will have to obtain all quantum-gravitational contributions to its path from 'the rest of the universe', and then it will move like a fish from a school of fish. Most importantly, such quasi-local fish will always have strictly positive inertial mass.

(On March 27, 2007, Prof. Warren W. Johnson, LSU, wrote: "Ah ha, caught you lying! You do have a radically different "theory" to compete with Einstein's theories." But LIGO fellow Warren W. Johnson is wrong. I strictly follow Einstein's GR, and am trying to remove all "miracles" in GR (resembling the projection postulate in QM), which preclude us from understanding the geodesic hypothesis, as explained by Alan Rendall. If you agree with Warren Johnson, or trust Chris Isham, try to uncover some "total field of as yet unknown structure", and send your proposal to Alan Rendall. I hope he will then re-write his online article.)

Notice that the vanishing property of tµv (M. Montesinos) is manifestation of the so-called ‘problem of time’: nothing can possible "move" in such block universe; cf. G.F.R. Ellis below.

My suggestion is to zoom on the "infinitesimal variables" [Ref. 3] and reveal the interplay of matter and geometry -- their bi-directional "talk" on the interface of the Archimedean (material) and non-Archimedean (geometrical) realms. The end result is a perfect continuum in the local mode of spacetime.

NB: This can only happen if there is a physical mechanism producing such perfectly smooth spacetime manifold, based on the so-called speed of light: the "duration" of the bi-directional "talk", in the local mode of spacetime, is zero.

This is the meaning of the statement 'Your Global Time is ZERO'. More in my talk on Wednesday, 25 November 2015. My first talk didn't attract the attention of the theoretical physics community, but once the "enhanced" and "advanced" LIGO fail miserably by November 2015, I hope people will get serious about GR:

"The representation of matter by a tensor was only a fill-in to make it possible to do something temporarily, a wooden nose in a snowman." (Albert Einstein's Last Lecture, April 14, 1954)

"The search for the gravitational energy-momentum tensor is often qualified as an attempt of looking for “the right answer to the wrong question”. [cf. C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation (W. H. Freeman and Company, New York, 1973), p. 467 - D.C.] ..."In traditional field theories, one arrives, after some work, at the energy-momentum object which is: 1) derivable from the Lagrangian in a regular prescribed way, 2) a tensor under arbitrary coordinate transformations, 3) symmetric in its components, 4) conserved due to the equations of motion obtained from the same Lagrangian, 5) free of the second (highest) derivatives of the field variables, and 6) is unique up to trivial modifications not containing the field variables. There is nothing else, in addition to these 6 conditions, that we could demand from an acceptable energy-momentum object, both on physical and mathematical grounds...."In the geometrical formulation of the general relativity, the components gmn(xa) play a dual role. From one side they are components of the metric tensor, from the other side they are considered gravitational field variables. If one insists on the proposition that “gravity is geometry” and “geometry is gravity”, then, indeed, it is impossible to derive from the Hilbert-Einstein Lagrangian something reasonable, satisfying the 6 conditions listed above."

"There had been a plot not to agree on an object to be guessed, but that each person, when asked, must give a truthful answer concerning some real object that was in his mind, and which was consistent with all the answers that had gone before. With only one question left, John Wheeler guessed: "Is it a cloud?" The answer was "Yes!"

Comment: Eric Schechter wrote that "if you take a medium-sized number and divide it by an enormous number, you get a number very close to 0."

Since the notions of infinity and infinitesimal are, in some (yet to be explained) sense, reciprocal, the latter can be illustrated with the following expression (notice that this is just an illustration of the puzzle stressed by Lucretius):

1/∞ → 0

We take a medium-sized number, 1 , to represent a finiteArchimedean thing (e.g., one meter or one second), and divide it by ... what kind of infinity? Potential infinity or completed/actual infinity? No matter what we choose, we cannot recover the finite Archimedean thing by multiplying "zero" by "infinity". We are "bartenders" (recall Thompson's lamp paradox).

That's the puzzle of the finiteArchimedean entities called 'space' and 'time' (local mode). Viewed from the local mode of spacetime, the infinitesimal -- the atom of geometry -- is the instantaneous state of Zeno's arrow. It is in 'absolute rest' with respect to all relativistic systems, hence we can define the elementary increment of physical variables (ds and dt) with respect to such "ether". I call it 'geometrical point', and stress that it is a non-Archimedean entity (global mode of spacetime), which builds up 'the grin of the can without the cat' (Alice), in line with the Continuum Hypothesis (CH). The latter is neither provable nor disprovable -- cf. Kurt Gödel. Why? Because the continuumemerges from the non-Archimedean realm of 'the universe as ONE', in which our mundane notions of "zero" and "infinity" do not hold anymore. They are simply not-applicable.

All we can say is that, depending on the "direction" we look at 'the universe as ONE', it looks like either infinitely small or infinitely large, as it wraps up the whole Archimedean 3-D space of present-day GR. Hence we can enjoy “self-contained” isolated systems, the asymptotic spacelike regime included (Adam Helfer).

The recipe is simple and unique. How else can you remove the jejune poetry in mathematical GR and differential geometry textbooks, encoded in expressions like "sufficiently small" and "smooth" (Piotr Chrusciel), and in stipulations that the Hausdorff topological space has somehow been made "connected" (Chris Isham)? There is no matter at the primordial level of 'pure geometry' to enable such "connection", which would show up as the affine connection (Graham Nerlich).

Thus, the event manifolditselfshould be endowed from the outset with the flexibility to be modified at each and every next event from 'the world lines of galaxies' (Weyl’s principle). Precisely what is implied by such flexibility?

To cut the long story short, the ‘no-prior-geometry’ demand fathered GR (MTW, p. 431), but by doing so it also fathered a century of confusion. No aspect of the geometry of spacetime should be "fixed immutably", i.e., "cannot be changed by changing the distribution of gravitating sources" (MTW, p. 429). Any fixed background, such as the topology of space (not determined in current GR) and the fixed relations of 'inside' vs. 'outside' in 3-D space (local mode), must be made (i) dynamical and (ii) totally removable in the global mode of spacetime. And that's why we need the global mode of spacetime: the "dark gaps" from the global mode are rendered/reduced to zero in the local mode, by the Arrow of Space and the so-called 'speed of light'. The physical/observable result is a perfect continuum, at all length scales. We don't have any other choice.

As the old saying goes, you pays your money and you takes your choice; but assume well-defined concepts of (i) energy density in GR and (ii) trajectory of quantum particles (quantum flexibility, not "fluctuations"), your choices narrow greatly. Regarding (ii), all particles simultaneously explore all potential paths (“smells all the paths in the neighborhood”, The Feynman Lectures on Physics, Vol. II, Ch. 19, The Principle of Least Action) before (=global mode) they jointly make the elementary (dt & ds) step along their trajectories; hence each and every step is unique and irreversible in the Heraclitean (non-Archimedean) time.

As to (i), the energy density is fixed both 'at a point' and 'viewed from infinity'. It's a package, and we can get it only with the global mode of spacetime in which the notions of 'geometrical point' and 'actual infinity' refer to ONE entity. Depending on the "direction" we look at 'the universe as ONE' from within the 3-D space (local mode), it looks either as 'infinitesimal geometrical "point" tending asymptotically toward zero', or as 'infinitely large and "expanding" volume of 3-D space'.

There is no other choice for quantum gravity. We first have to fix the long-standing problems of QM and GR, and then all pieces of the jigsaw puzzle will snap to their places -- effortlessly. Then the proper math will also show up -- effortlessly. I am sure professional mathematicians will understand what I mean.

If you agree with Chris Isham, you'll be playing with the drawing below forever.

"(W)hich physical assumptions are being tacitly made whenever one postulates the existence of a Lorentzian metric? Only after these assumptions are exhibited can one start to systematically relax them; thus, answers to the above question may be helpful for the study of more general space-time structures.

"Primitive concepts are taken to be events, counting of events, causal relationships and the ability to compare measurements; the corresponding mathematical structures are a differentiable manifold, volume element, causal vectors and affine connection(s), leading to the notion of an 'event manifold'.

"The key assumption, which I will call 'deformability', is that the event manifold's physical structure is allowed to vary freely.

"The proof of the Weyl-Cartan theorem is then reviewed to establish the result that any deformable event manifold must be Lorentzian.

2 Event Manifolds

"I assume that space-time is a connected n-dimensional differentiable manifold M. At x E M, local measurements (e.g., evaluating vector fields) are performed using a basis of the tangent space TxM. In order to have a means to compare local measurements at different points, I require the manifold to be endowed with an affine connection.

"So far my considerations have been very general, and the symmetry group G is by no means uniquely determined. Only now the key idea of General Relativity comes into play: rather than being fixed as in Newtonian theory, the local physical structure on the space-time manifold is itself a variable; it depends on the distribution of matter in the universe (and on boundary conditions)."

p. 25: "Space and time are such crucial categories for thinking about, and describing, the empirical world, that it is bound to be ferociously
difficult to understand their emerging, or even some aspects of them emerging, from 'something else'......p. 46: "As we said in Section 1, we intend ‘the emergence of time’ to also cover the emergence of spacetime, and so space."

You acknowledged that a future directed, time-like unit vector field, for which no natural choice exists in general, is characterized indirectly and becomes explicitly available only after solving the equations (arXiv:0903.5160v1 [gr-qc], p. 17).

I've been trying to argue that this problem can only be solved by recovering the reference fluid in GR.[snip]

It seems to me -- please correct me if I got it wrong -- that Eqs 1 and 2, pp. 35-36 at the link above, may offer a solution to the ‘most embarrassing observation in physics’ (Ed Witten) and explanation of the apparent finite age of the universe:

"WMAP definitively determined the age of the universe to be 13.73 billion years old to within 1% (0.12 billion years) - as recognized in the Guinness Book of World Records!"

The model suggests a dual age of the universe, such that the evaluation of the statement from WMAP Team will be, in German, YAIN (both yes and no).

Your critical comments will be appreciated, and will be kept private and confidential.

You and Dr. Niedner wrote: "It is an example of an everywhere continuous but nowhere differentiable curve. We can construct the Koch curve as a final product of an infinite sequence of steps. At each step, the middle third of each interval is replaced by the other two sides of an equilateral triangle."

Suppose we consider, as a Gedankenexperiment, an infinite sequence of steps, and assume "the presence of a minimal length" (p. 5), such that (operational definition) the initial 'interval' in Fig. 1 at this "minimal length" becomes _sufficiently small_ (the key expression from the Equivalence Principle in GR textbooks), hence can be considered as 'infinitesimal point' -- the very same infinitesimal point "inside" which the state of the Thompson's Lamp becomes _indecisive_ . Namely, a superposition of |on> and |off> state(s), which can never be "collapsed" bzw. observed.

We would have an ultimate cutoff by such (Planckian?) 'sufficiently small minimal length', yet we won't be able to compute the actual length of the whole Koch curve (nor the Hausdorff dimension), because the _sufficiently small_ infinitesimal length will act as 'numerically finite but physically unattainable boundary/cutoff'.Stated differently, the final curve won't be "infinitely long", as you put it, but _indecisive_ .

I believe the implications for the notion of "delocalization of point like objects" (p. 3), as well as for those depicted in Fig. 4, are obvious, but let me first state my question:

> as far as I understand, I would say that in some sense your comment is> correct. the conventional computation of the Hausdorff length is modified> as we showed in our paper.

With the Thompson's Lamp Paradox, I think the situation changes drastically.

> The introduction of a length scale breaks the self similarity property> of the erratic path of a quantum particle.

Please see above.

> However some points don't seem to be correct in your comments. In> particular the reference to the infinitely long case, which is the> conventional case rather than that in the presence of a cut off as shown> in Eq. 16.

Perhaps you didn't have time to consider my Gedankenexperiment vs. yours (the alleged cutoff in Eq. 16).

Note: The introduction of a minimum length scale as a 'numerically finite but physically unattainable boundary/cutoff' follows from the non-Archimedean nature of the geometric realm of 'the grin of the cat without the cat' (Alice): we cannot reach the "hidden unmoved mover" (Karel Kuchar) and the Aristotelian First Cause from within the local mode of spacetime. Corollary: speculations about some dirty black hole thermodynamics, noncommutative gravity, non-commutative micro black holes, entropic "force" (the latter translates to 'information force'), etc., are unjustified.

"The time at which wave-function collapse occurs is ill defined in relativity; however the simultaneity associated with the shape-dynamic CMC foliation could bring interpretational clarity. This is also true for the ‘problem of time’ [14], which arises from the ambiguity in the time evolution in superspace if foliation invariance (many-fingered time) is made inviolate."

May I request some interpretational clarity: please tell me how you plan to resolve the quasi-local mess in GR,

If your colleagues have some insights, I will appreciate their feedback as well.

Regards,

Dimi-----------

Note: The only way I could think of "the time at which wave-function collapse occurs" is by speculating about Weyl's Principle being applied to some preferred foliation of spacetime into 3-D spacelike hypersurfaces, to fix the temporal order for spacelike separated "points" -- simultaneity-at-a-distance -- hence 'simultaneity surfaces' along which quasi-local effects propagate among the "school of fish", in which every "fish" follows pre-correlatedquasi-local geodesics.

The present-day GR explicitly forbids such luxury, as well as some 'nondynamical time parameter' (Unruh & Wald).

Niall Murchadha and Julian Barbour claim that have found "the configuration space of general relativity" that "could bring interpretational clarity", and I offered them 'the test of the pudding'. Are they implying some null surfaces backward in time to resolve the quasi-local mess in GR? Check out the 1982 paper by R. Penrose; the problem has been agonizingly clear since the inception of GR.

Apart from that, Niall Murchadha and Julian Barbour have brains, which could not have worked in any "timeless world" from GR, nor with some 'time variable' in STR, used to explain the energy conservation during that 'time variable', after Noether's Theorem.

The human brain needs an arrow of spacetime. Otherwise the human mind must act on its brain, which is sheer parapsychology. The latter has been explored by many people since the Roman Empire, with the same dead-end result.

If you seriously believe that the flow of spacetime is within the framework of the theory of relativity "an illusion", you have a choice: fix the quantum theory and theory of relativity by incorporating its "dark energy", or do parapsychology.

Or simply ignore this web site, and pretend that you've never learned anything from it, like Julian Barbour does.

Students are kids, and I think nobody can blame them for being unable to grasp QM, given the fact that their tutors and professors are profoundly confused in the first place ("large variation in faculty views on many topics in QM", as you mildly put it).

May I elaborate on John Cramer's idea (p. 4): "repeats until" ... at which point the transaction is completed, and the observer can finally witness the *already*-completed transaction (post factum). I deliberately use temporal notions to show that (i) we're dealing with two kinds of time, and (ii) this language doesn't work. Perhaps if we allow these two kinds of time to co-exist peacefully, the crux of QM can be resolved:

p. 2: "...the process of electromagnetic radiation should be thought of as an interaction between a source and an absorber rather than as an independent elementary process. (footnote 1)"

Footnote 1, p. 3: "... it is as absurd to think of light emitted by one atom regardless of the existence of a receiving atom, as it would be to think of an atom absorbing light without the existence of light to be absorbed."

p. 3: "... an advanced incoming field that is present at the source simultaneous with the moment of emission. The claim is that this advanced field exerts a finite force on the source which has exactly the required magnitude and direction to account for the observed energy transferred from source to absorber; ..."

p. 3: "The crucial point to note about the Wheeler-Feynman scheme is that due to the advanced field of the absorber, the radiative damping field is present at the source at exactly the time of the initial acceleration. Quite simply, if a retarded electromagnetic disturbance propagates for a time tbefore meeting the absorber then the absorber will be a distance ct from the source. The advanced field propagates with the same speed c across the same distance and thus will arrive at the source exactly time tbefore the absorber field is generated, i.e. at the time of the initial acceleration."

p. 4: "The transaction is completed with a “handshake”: the offer and confirmation waves combine to form a four dimensional standing wave between emitter and absorber. (...) Any observer who witnesses this process wouldperceive only the completed transaction, which would be interpreted (post factum - D.C.) as the passage of a particle (e.g. a photon) between emitter and absorber."

p. 5: "The process is atemporal and the only observables come from the superposition of all “steps” to form the final transaction. (1986, fn. 14, p. 661)"

In the late 1950's, Ernst Specker posed the question of whether the omniscience (total knowledge) of God extends to events which would have occurred if something had happened, which did not, in fact, happen (Ernst Specker, “Die Logik nicht gleichzeitig entscheidbarer Aussagen,” Dialectica 14, 239-246 (1960); p. 243).

This leads to the questions of 'the universal truth function', and "why does quantum theory not have this sort of complementarity" [Ref. 1].

My scattered thoughts on KS Theorem and truth evaluation in Quantum Theory can be read at

Regrettably, I cannot attend the Colloquium in Honour of Ernst Specker, and can only wish him all the best upon his Birthday 'by distance', although I will be, in some sense, present as well, since we are all entangled.

May I take this opportunity to invite you at my talk on quantum gravity,

p. 3: "(O)ne must imagine that the outcome of a measurement (or equivalently, the property that is measured) is context-dependent — whether a gem is seen or not in the first box depends on whether that box was opened together with the second or together with the third......"To get this kind of contextuality, it is necessary to find a situation wherein there are very specific sorts of limitations on joint measurability — there must exist a triple of measurements that can only be implemented jointly in pairs. For projective measurements in quantum theory, this sort of limitation on joint measurability does not occur.......pp. 27-28: "Specker’s parable provides an interesting new kind of foil, because the kind of complementarity it exhibits — three measurements that can be implemented jointly pairwise but not triplewise – is something that is not found among projective measurements in quantum theory. This prompts the question: why does quantum theory not have this sort of complementarity?"---------

Note: Just a hint to the question above: suppose the {1,0,1} rule was not broken, and the full catalogue of expectation values for the spin-1 system were present (no "uncolored" section from KS sphere). Then people could develop a relativistic QM that could explain the "collapse" with some "hidden" stuff, and prove Schrödinger wrong, namely, 'a variable would have a definite value before I measure it; then measuring it simply means ascertaining the value that it has.'

"Let me say at the outset, that in this discourse, I am opposing not a few special statements of quantum mechanics held today (1950s), I am opposing as it were the whole of it, I am opposing its basic views that have been shaped 25 years ago, when Max Born put forward his probability interpretation, which was accepted by almost everybody....."I don't like it, and I'm sorry I ever had anything to do with it."

Suppose you are an Eskimo, and you have never seen, and will never see an elephant in your whole life. Yet you can nevertheless make observations on elephant's trunk by means of two "complementary" devices, 'nose' and 'arm'. You know that Heisenberg relations preclude you from observing simultaneously the "nose" and the "arm" of elephant's trunk (the position and momentum of an electron, say). Then you're struck by Schrödinger's 1935 paper:

"In general, a variable has no definite value before I measure it; then measuring it does not mean ascertaining the value that it has. But then what does it mean?"

It means you cannot observe elephant's trunk with (inanimate) devices at the length scale of tables and chairs. It does not mean that the trunk doesn't exist. Fortunately, the Kochen-Specker Theorem can help you understand the true, UNdecidable quantum "trunk"; check it out with your own brain here. Richard Feynman claimed (12 August 1983): "We can do the arithmetic, but we cannot picture the car!" Yes we can. All we need is a brain.

Carsten Held stated that the KS theorem, "by its mathematical nature, is not empirically testable", and since Ronnie Hermens (and Wiki) seem to have some troubles with it, may I offer a brief, personal, and biased interpretation, ensuing from Ernst Specker's tripod.

Ernst Specker was eager to clarify the answer to a truly fundamental question: is it possible to distillall conceivable quantum states as 'observables in 3-D space', such that (i) there will be no counterfactuals (cf. Karl Svozil) and (ii) the unitarity principle will be uphold, that is, the probabilities for all conceivable quantum observables will sum up to unity. If that were possible, one could make a "reverse engineering" of a quantum system, by exposing all of its localizable observables, after which the cornerstone questions of Quantum Mechanics (QM), posed by Erwin Schrödinger in 1935 and in November 1950 (cf. below), would have acquired a dead simple answer: hidden variables. Then QM would be just a statistical theory based on the unitarity principle, and will also be marred with the intractable parapsychology of those "hidden variables" (maybe sub-quantum, or maybe noncontextual, but you never know, because all this parapsychological stuff is "hidden" from the outset). Thank God, the Kochen-Specker Theorem proved this whole mess wrong. But it also showed that the notion of 'complete set of observables pertaining to a quantum system' is far more subtle and rich than expected, due to the presence of an UNdecidable KS state that does not belong to this set.

Imagine three quantum guys, Tom, Dick, and Harry, and think of the "spectral decomposition" of their quantum states as being presented by their hands, with the following specifications: upon observation at particular instant, they all have to simultaneously raise their hands (hermitian operators). Thus, each of them can, and have to, raise either his left hand (L), or right hand (R), or both (B). Recall that, unlike probabilities in classical statistical physics, probabilities in QM originate in Pythagoras' theorem in n dimensions (C. Isham, Lectures on Quantum Theory, Sec. 2.1.2, get pp. 16-17 from here), so if one of the quantum guys cannot have orthogonal "states" for his hands, he will ruin the whole system.

The famous KS Theorem (download an explanation by R.I.G. Hughes from here) says that the (spin 1) system {Tom_Dick_Harry} will exhibit the following "paradoxical" (from the viewpoint of classical physics) situation: if Tom and Dick happen to possess context-dependent and well-defined ("an unequivocaltrue-false value", cf. Isham and Butterfield above) quantum states of their hands (either L, or R, or B), then in that same instant Harry will not have any hands at all. For if the poor guy had 'hands' in that same instant, he would have the opportunity to choose one of his context-dependant quantum states -- either L, or R, or B. He will instead be shifted to the "uncolored" section from KS sphere.

Next time you "measure" the system {Tom_Dick_Harry}, exactly the same thing can happen to Dick. Or to Tom. So, any time you observe some classical presentation of 'quantum state', be aware that nevertheless something essential is missing. It showed up in the case with Harry, yet it "covers" the whole system {Tom_Dick_Harry}, much like the "dark" energy from 'empty space'. Poetically speaking, the case with Harry was the "revenge" of the quantum Noumenon pertaining to {Tom_Dick_Harry} for our efforts to push it at the length scale of (unanimated) tables and chairs. Unlike the measuring devices examined in QM textbooks, the human brain can operate with its presentation of 'potential reality' elevated at the length scale of tables and chairs, as demonstrated with the example of three "measurements" (cf. the three sayings above). If our brains can operate with their UNspeakable potential reality, so can 'the universe as a brain'.

Notice that if we constrain Tom, Dick, and Harry to raise only one hand (either L or R), people would entertain "quantum computing" and "topological quantum computation" (e.g., Michael H. Freedman), because the implications from KS Theorem will be obscured. But as the UNdecidable quantum state is still working in the quantum world, it cannot be harnessed with unanimated devices.

I employ the notion of 'potential reality' to signify the UNdecidable KS quantum state -- the genuine 'quantum reality out there'. Can't fit it in any Hilbert space whatsoever. If you prefer, think of it as Reichenbach's Common Cause Principle.

In summary, the Kochen-Specker Theorem demonstrates the UNdecidable KS quantum state. The latter is far more profound and important than the contextuality alone [Ref. 2] ("not all observables can be assigned definite values that are independent of the measuring context", cf. Ronnie Hermens).

Namely, the conclusion that "onlycontextual values can be ascribed" [Ref. 2] does not shed light on the implications from the fact that contextuality is invariant to the particular place of the observables in the 3x3 array. As I wrote above, the same thing that happened to Harry can happen to Dick, or to Tom.

Nothing -- not even some "contextuality" -- can save Harry (or Dick, or Tom) from the case in which he (or Dick, or Tom) must not possess any hands.

If at some instant of measurement Harry must not have any classical hands whatsoever, yet at some other instant he -- the same Harry -- can and will obtain some contextual, well-defined, classical-able hands, then we are facing a phenomenon far more important than "contextual hands" alone. Namely, we can tell apart the contextualized classical-able hands, which can be filtered through the classical spacetime as 'classical hands', from those "intact" quantum states that can't. Which does not imply that the latter do not exist. In my view, only some quantum Noumenon can secure the ontological contextuality of quantum objects. The ontological contextualitymay or may not provide 'the same Harry' with contextualized classical-able hands. It is also UNdecidable and Unspeakable, and of course 'not empirically testable'. Thus, if you observe the "hands" of some quantum object, be aware that you're only observing its contextualized classical shadows on Plato's cave, emanating from the ultimate quantum potential reality.

Three years ago, I explained to my teenage daughter a similar puzzle in GR, regarding the Hole Argument. With slight modifications, the corresponding 'KS Theorem for teenage girls' would be as follows. Imagine you cannot see your finger nails "bare", without nail varnish. One way to think of such peculiar situation is that your 'bare finger nails' do in fact exist, but are somehow banned from showing up to you. Namely, your 'bare finger nails' can show up to you only after you cover them with your nail varnish, with different colors depending on your mood, but then they won't be 'bare' anymore. They will be "contextualzed", with particular color. Well, KS Theorem (roughly) says that (i) if you use three different nail-varnish colors for each nail, and (ii) if you have to paint three (or more) of your nails, then one of the them would have to show to you its "bare" nature: the UNdecidable KS state. Which would, in turn, ruin your whole manicure. To avoid such disasters, you should use maximum two nail-varnish colors, and then all your "bare" finger nails will be safely covered with some particular "contextualzed" color. The downside of such (seemingly perfect) manicure will be that you may never understand QM nor GR. Capiche?

NB: If the reader wishes to refute the "adult" interpretation of KS Theorem above, please start with converting the "uncolored" section from KS sphere to some Hilbert space (your choice), to match the case in which Harry would obtain some contextual, well-defined, classical hands, at the expense of either Dick or Tom being shifted to the same "uncolored" section from KS sphere. What is the 'time parameter' pertaining to the dynamics of the reversible (KS sphere <--> Hilbert space) transitions, for all observables from {Tom_Dick_Harry}? Please don't hesitate!

In a drastic contrast to the "collapse" and the Eigenvalue-Eigenstate Link from the old Copenhagen School [Ref. 1], the PR interpretation of QM employs the phenomenon of 'emergence' (e.g., Isham and Butterfield): in the case with {Tom_Dick_Harry}, one could only observe, at particular instant of time, the emergence of only one of its latent observables, say, {Tom, R}, just like the end result from the correlation and "negotiation" (global mode of spacetime) that led finally to John Wheeler's 'cloud'. This one final result (one-at-a-time) is 'physical reality' (local mode of spacetime), while the rest of Tom's latent states, plus Dick's latent states, plus Harry's UNdecidable KS state constitute the 'potential reality' of the system {Tom, Dick, and Harry}, which is in turn rooted on the quantum Noumenon -- "The ideal monad has no windows" (Döring and Isham). This is entirely different from any 'modal' or 'contextual hidden variable' theory.

It goes without saying that a rigorous presentation of the ideas in the paragraph above is not available. These are just ideas presented with words. Back in 1935, Erwin Schrödinger also offered some very general ideas, presented with words:

"The rejection of realism has logical consequences. In general, a variable has no definite value before I measure it; then measuring it does not mean ascertaining the value that it has."

The second part from the last sentence was totally forgotten by the mainstream theoretical physics community, and only Henry Margenau paid attention to it. Yet even today very few QM textbooks mention KS Theorem, which is rooted on this forgotten (or rather ignored) consideration spelled out by Schrödinger in 1935. As to the first part, "a variable hasno definite value before I measure it", perhaps the intact quantum world 'out there' exists as the UNdecidable quantum state.

If this is the case chosen by Mother Nature, the mysterious transition from quantum to classical, which is essential to QM textbooks [Ref. 3], should be explained with the "back bone" of the quantum world -- potential reality.

Notice that the quantum truth functional in QM textbooks (e.g., Ref. 4, p. 314) is not applicable to 'potential reality' due to the presence of the generic "intact" (not limited to KS Theorem) UNdecidable quantum state.

Of course not. The whole point is that there must exist something that keeps the "sameness" (Genidentität, Kurt Lewin) of thisspin-half particle, so that it can "pass through" it (the "intact" UNdecidable quantum state, or 'bare finger nails'), and be able to switch between its allowed states, in line with the conservation law known from QM textbooks.

Can we, with our Boolean logic, think of a quantum particle as possessing simultaneously perfectly well-defined position and momentum (ibid., p. 314)? Of course not. Does that mean that we can "impose bans" on such UNdecidable quantum state in the quantum world, just because we cannot think about it? Of course not. The "non-commutative" quantum state is simply notpoint-like. It may be perfectly well-defined as 'potential reality', yet not point-like. It has to be point-like only in classical mechanics, like the point-like states of a Frisbee along its classical trajectory. (The hypothetical abilities of the human brain to observe and act on the UNdecidable quantum vacuum state requires special considerations.)

Let's not mix apples with oranges, because the 'time' in the quantum world (no time operators in QM) is not like the one at the classical world, and the "intact" UNdecidable quantum state simply cannot get there. With unanimated measuring devices, we can "measure" only one frozen point from the anti-relativistic "time parameter" in the Schrödinger equation, and by imposing such wrong, albeit inevitable, classical filter on the quantum world we may develop distorted and misleading impression about some "time parameter in the Schrödinger equation". The experimental fact that even by detecting electrons one-at-a-time they nevertheless build an interference pattern [Ref. 5] demonstrates their quasi-local nature, yet the probability for detecting individual electrons on the screen, as point-like events, is blind and deaf to their quasi-local nature: the interference pattern can only be produced if the individual electron was able to "sense" the two slits simultaneously, while the probability for its detection on the screen refers to registered events that are inevitably point-like facts.

This should be the starting point for explaining "the central mystery of quantum mechanics" (Richard Feynman): the nature of the quasi-local UNdecidable quantum state. It is not a 'fact', and cannot be presented with any probabilities, as was the case with Harry above. If you teach Quantum Mechanics, start with the double-slit experiments and finish with the KS Theorem, which is grounded on the statement made by Erwin Schrödinger in November 1950:

“It seems to me that the concept of probability is terribly mishandled these days. Probability surely has as its substance a statement as to whether something is or is not the case — an uncertain statement, to be sure. But nevertheless it has meaning only if one is indeed convinced that the something in question quite definitely is or is not the case. A probabilistic assertion presupposes the full reality of its subject.”

NB: There is nothing "probabilistic" or "stochastic" in the quasi-local UNdecidable quantum state that neither 'is' nor 'is not'. Can't fit it in any Hilbert space.

There are three totally different cases in QM. With just one degree of freedom, we (not Mother Nature) calculate an observable with a single eigenvalue ("an eigenstate of k belonging to an eigenvalue k'," P.A.M. Dirac, p. 154); the second case is confined to 'contextual values' (we constrain Tom, Dick, and Harry to raise only one hand -- either L or R; see above); and the third case deals with the absence of any classical hands whatsoever, after KS Theorem and the Free Will Theorem.

Quantum Mechanics is about this third case. Can't fit it in any Hilbert space.

Surely the quasi-localUNdecidable quantum state cannot be filtered through the spacetime of STR and the Boolean logic of propositions: the truth evaluator will be YAIN (both yes and no). Which is entirely different from the "toposification" of quantum theory suggested by Chris Isham -- neither true nor false, but "somewhere in between" [Ref. 6].

To quote Niels Bohr, "Mathematical clarity has in itself no virtue. A complete physical explanation should absolutely precede the mathematical formulation." And the physical explanation has been spelled out by Schrödinger and Margenau. Only the math is unknown.

To move further (Machian quantum gravity), try to unravel the origin and mechanism of instantaneous inertial reaction "forces", starting solely from Albert Einstein's postulate that "the metrical character (curvature) of the four-dimensional spacetime continuum is determined at every point by the matter there, together with its state" (Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie, 1917).

No way. Something inherently quasi-local is again missing (try an experiment with your brain here), as hinted by KS Theorem and CK Free Will Theorem. In the case of GR, the torsional degrees of freedom are excluded from the outset. But I'll stop here, because, as many people complain, the story will (again) become deadly boring.

"Thus quantum mechanics occupies a very unusual place among physical theories: it contains
classical mechanics as a limiting case, yet at the same time it requires this limiting case for its own formulation."

p. 190: "... when constructing a quantum description of a physical system it is necessary to restrict oneself to a single framework, or at least not mix results from incompatible frameworks.

p. 194: "One cannot use a single framework to answer all possible questions about a quantum system, because answering one question will require the use of a framework that is incompatible with another framework needed to address some other question.

p. 195: "If two or more frameworks are compatible, there is nothing problematical in supposing that the corresponding conclusions apply simultaneously to the same physical system. (...) Consequently one can think of F1, F2, ... as representing alternative "views" or "perspectives" of the same physical system, much as one can view an object, such as a teacup, from various different angles. Certain details are visible from one perspective and others from a different perspective, but there is no problem in supposing that they all from part of a single correct description, or that they are all simultaneously true, for the object inquestion.

p. 196: "Conceptual difficulties arise, however, when two or more frameworks are incompatible. (...) ... just as there is no property corresponding to Sz = +1/2 AND Sx = +1/2 for a spin-half particle. (...)

"Incompatible frameworks do give rise to conceptual problems when one tries to apply them to the same system during the same time interval. (...) The difficulty comes about when one wants to think of the resultsobtained using incompatible frameworks as all referring simultaneously to the same physical system, or tries to combine the results of reasoning based upon incompatible frameworks. It is this which is forbidden by thesingle framework rule of quantum reasoning."------

p. 261: "Each of the nine observables in (22.14) commutes with four others: two in the same row, and two in the same column. However, it does not commute with the other four observables. Hence there is no reason to expect that a single value functional can assign sensible values to all nine, and indeed it cannot.

"The motivation for thinking that such a function might exist comes from the analogy provided by classicalmechanics, as noted in Sec. 22.1. What the two-spin paradox shows is that at least in this respect there is a profound difference between quantum and classical physics.

p. 263: "The nonexistence of universal quantum truth functionals is not very surprising. It is simplyanother manifestation of the fact that quantum incompatibility makes it impossible to extend certainideas associated with the classical notion of truth into the quantum domain. Similar problemswere discussed earlier in Sec. 4.6 in connection with incompatible properties, and in Sec. 16.4 inconnection with incompatible frameworks.

p. 268: "Note that quantum truth functionals form a perfectly valid procedure for analyzing histories (and properties at a single time) as long as one restricts one's attention to a single framework, a single consistent family. With this restriction, quantum truth as it is embodied in a truth functional behaves in much the same way as classical truth. It is only when one tries to extend this concept of truth to somethingwhich applies simultaneously to different incompatible frameworks that problems arise.

p. 314: "For the purposes of the following discussion it is convenient to refer to the idea that thereexists a unique exhaustive description as the principle of unicity, or simply unicity. This principleimplies that every conceivable property of a particular physical system will be either true or false,since it either is or is not contained in, or implied by the unique exhaustive description. Thusunicity implies the existence of a universal truth functional as defined in Sec. 22.4.

"But as was pointed out in that section, there cannot be a universal truth functional for a quantum Hilbertspace of dimension greater than two. This is one of several ways of seeing that quantum theory isinconsistent with the principle of unicity, so that unicity is not part of quantum reality."

-----

p. 375:

(Note: Bob Griffiths didn't comment on the Kochen-Specker diagram (see the 17 projections, p. 70, in S. Kochen and E.P. Specker, The problem of hidden variables in quantum mechanics, J. Math. Mech. 17 (1967) 59-87), but instead suggested "the helpful discussion of these and other results" offered by N. David Mermin. Recall that Kochen and Specker used a finite set of spin-1 observables (not spin-2, as in the case examined by Mermin). The essence of ‘the Kochen-Specker Theorem’ is the argument for nonexistence of truth functions on finite sets of projections -- nothing to do with Bell's 1966 paper (cf. p. 452, footnote 19). Bob Griffiths is to be taken with a grain of salt. The same, of course, applies to my interpretation. D.C.)

p. 4: "A remark on assumptions. In the previous section, we supposed the measured observables to be conserved. This is necessary to carelessly time-translate the projection operators. The translated observable here is the idler photon measurement. This determines the beam of photon (so its momentum) and is clearly conserved."

p. 2: "The second observation that I wish to report here is that the collapse of a wave function is associated with the collapse of the energy-momentum tensor. Since it is the energy-momentum tensor that determines the spacetime metric, the position measurements alter the spacetime metric in a fundamental and unavoidable manner. Therefore, in the absence of external gravitating sources (which otherwise dominate the spacetime metric), it matters, in principle, in what order we make position measurements of particles [D.V. Ahluwalia, Quantum Measurement, Gravitation, and Locality, gr-qc/9308007]. Quantum mechanics and gravity intermingle in such a manner as to make position measurements non-commutative. This then brings to our attention another intrinsic element of gravity in the quantum realm, the element of non-locality."

pp. 8-9: "The result indicates that there is no evidence so far of any quantum nature of spacetime above the Planck length. Spacetime there (distance of 7.3 billion light years from Earth - D.C.) is smooth and continuous."

[Ref. 6] C J Isham, Is it true; or is it false; or somewhere in between? The logic of quantum theory, Contempory Phys., 46(3), 207-219 (2005)

N. Gisin: "Finally, one should mention that a way out of our entire argumentation is to assume the existence of one preferred universal reference frame which determines unequivocally one and only one time ordering for all events."

Welcome aboard, Nicolas Gisin. You just discovered John Bell [Ref. 1] and the legacy of Schrödinger and Margenau,

D. Chakalov----[Ref. 1] J. S. Bell, in The Ghost in the Atom: A Discussion of the Mysteries of Quantum Physics, ed. by P.C.W. Davies and Julian Russell Brown, Cambridge University Press, 1993, pp. 49-50.

“The reason I want to go back to the idea of an aether here is because in these EPR experiments there is the suggestion that behind the scenes something is going faster than light. Now, if all Lorentz frames are equivalent, that also means that things can go backward in time. [This] introduces great problems, paradoxes of causality and so on. And so, it’s precisely to avoid these that I want to say there is a real causal sequence which is defined in the aether.”

Perhaps that full-time base-scientist (Josh should know his name) who was trying to understand and develop anti-gravity devices at Wright-Patterson Air Force Base [Ref. 1] will be interested to check out

Perhaps a Machian-type theory of gravity can explain the origin and mechanism of (instantaneous?) inertial reaction "forces" and their reversible cancellation, so that you can levitate your "anti-gravity device".

As to the McCarthy witch-hunt period [Ref. 1]: would you turn up your collar to read the facts about 9/11?

I wish you and your colleagues could at least elaborate on the origin and mechanism of inertial reaction "forces", but I'm afraid you won't do it either. You didn't even respond to my email regarding your lrr-2009-6 and the mythical "future conformal null infinity".

"I point out that Wright-Patterson Air Force Base provided financial support for the Syracuse and King's College groups (among several other relativity groups) from the mid 1950s to the early 1970s - during a most productive period. A question often asked is why did they do so. Though I was not privy to any internal Air Force information, once, when I spent a three month period working at the base, a full-time base-scientist remarked to me that they hoped to beable to understand and perhaps develop anti-gravity devices. It does seem likely that this idea played some role in their financial support........

"... a few weeks after I arrived in Syracuse I saw that a well-known left-wing journalist, I.F. Stone was giving a public talk. I went to the talk but with considerable trepidation since I had come from a fairly left-wing family background and the time was at the peak of the Joe McCarthy witch-hunt period. In my mild state of paranoia I actually had my collar turned up so that I would not be recognized."

Since you offered the idea that the universe may be "secretly acausal but only the universe knows about it", please consider the possibility that the universe may not be "acausal" but pre-correlated (Leibnitz) and bootstrapped (Geoffrey Chew): think globally, act locally. Perhaps all we need is to model the universe as a brain,

You also wrote: "To move forward, physics had to abandon an apparently ironclad piece of commonsense that “where there is a wave something must be waving.” I would not be at all surprised if it turns out that to move forward, we have to abandon an equally ironclad piece of commonsense. I leave it to the reader to identify that piece."

Have you seen a walking centipede? The legs exhibit a correlated wave pattern. Perhaps only the 'universe as ONE' (global mode of spacetime) can evoke quantum "waves" -- there is no source that is "waving", in QM textbooks.

As to GWs, "we have to abandon an equally ironclad piece of commonsense", as you put it. Please see the first link above.

I will highly appreciate your comments, as well as the professional feedback from your colleagues.

I read very carefully your papers on torsion degrees of freedom, but fond it impossible to hear and follow your talk on June 10th last year, PIRSA:09060061. If you have a written version, please send me the link.

May I ask a question. Regarding your latest arXiv:0911.1636v3 [gr-qc] and your Lecture Notes MP476: Cosmology (dynamics of length scale R, p. 43), I wonder how you would pinpoint some dynamics of the so-called "dark energy".

I look at my wristwatch and claim that there is a "global" cosmological time, read by my wristwatch, such that, say, in the past 5 min the global 'length scale' R has been "increased". With respect to what? Can you introduce some Akasha-like "memory" of the whole universe, such that some (global) observer could verify that some "increase" of R has indeed happened w.r.t. some earlier value of R five min ago?

I cannot imagine how anything could be "conserved" in the cosmological time driven by such dynamical "dark" energy of (whatever). Please advise.

Note: In one sentence: if something can be 'conserved in the time read by a clock', then it will be 'observable', and won't be "dark", because its source could be traced back and identified (cf. note 3). Again, the (global) time in which something can be 'conserved' is not the (local) time read by our mischievous clocks. It is the global mode of time produced by the dynamics of 3-D space itself (cf. Fig. 2 above). Any approach based on the Hamiltonian formulation of GR is inadequate from the outset, because the intrinsic dynamics of 3-D space, produced by its "dark" energy, cannot be captured within the 3-D space itself. You can never identify some isolated system in GR, and will always be haunted by problems at "infinity" (Paul Tod, 01:19-02:05):

To clarify the problems mentioned above, I will quote from a well-known text by S. Weinberg (emphasis added), in which the notion of 'torsion' has never been mentioned, and will offer my biased, personal, and frank comments.

Steven Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, Wiley, January 1972; ISBN: 978-0-471-92567-5.

p. 68: "Although inertial forces do not exactly cancel gravitational forces for freely falling systems in an inhomogeneous or time-dependent gravitational field, we can still expect an approximate cancellation if we restrict our attention to such a small region of space and time that the field changes very little over the region. Therefore we formulate the equivalence principle as the statement that at every space-time point in an arbitrary gravitational field it is possible to choose a "locally inertial coordinate system" such that, within a sufficiently small region of the point in question, the laws of nature take the same form as in unaccelerated Cartesian coordinate systems in the absence of gravitation.

"There is a little vagueness here about what we mean by "the same form as in unaccelerated Cartesian coordinate systems," so to avoid any possible ambiguity we can specify that by this we mean the form given to the laws of nature by special relativity, for example, such equations as (2.3.1), (2.7.6), (2.7.7), (2.7.9), and (2.8.7).

"There is also a question of how small is "sufficiently small." Roughly speaking, we mean that the region must be small enough so that the gravitational field is sensibly constant throughout it, but we cannot be more precise until we learn how to represent the gravitational field mathematically. (See the end of Section 4.1.)"

p. 93 (the end of Section 4.1): "There are in general many generally covariant equations that reduce to a given special-relativistic equation in the absence of gravitation. However, because we only apply the Principle of General Covariance on a small scale compared with the scale of the gravitational field, we usually expect that it is only g_mv and its first derivatives that enter our generally covariant equations. With this understanding we shall see in this and the next chapter that the Principle of General Covariance makes an unambiguous statement about the effects of gravitational fields on any system, or part of a system, that is sufficiently small."

Comment (D.C.): This isn't any mathematical treatment of "sufficiently small". This is pure poetry, spiced with math.

......

p. 106: "Write the appropriate special-relativistic equations that hold in the absence of gravitation, replace n_mv with g_mv and replace all derivatives with covariant derivatives. The resulting equations will be generally covariant and true in the absence of gravitation, and therefore, according to the Principle of General Covariance, they will be true in the presence of gravitational fields, provided always that we work on a space-time scale sufficiently small compared with the scale of the gravitational field."

Comment (D.C.): Such kind of thinking is typical to people like CEOFOP.

..........

p. 149: "Although a freely falling particle appears to be at rest in a coordinate frame falling with the particle, a pair of nearby freely falling particles will exhibit a relative motion (Sic! - D.C.) that can reveal the presence of a gravitational field to an observer that falls with them. This is of course not a violation of the Principle of Equivalence, because the effect of the right-hand side of (6.10.1) becomes negligible when the separation between particles is much less than the characteristic dimensions of the field."

"Thus we cannot learn anything about the time evolution of the gravitational field from the four equations {XXX} (7.5.1). Rather, these equations must be imposed as constraints on the initial data, ... ."

Comment (D.C.): Typical non sequitur. Besides, the Cauchy problem for the Einstein equations cannot be resolved: the global behavior of solutions of Einstein's equations cannot be uniquely defined (cf. Paul Tod's metaphysics at 01:19-02:05, and the discussion of angular momentum at 46:22-46:35).......

p. 166, Eq. (7.6.7): "the total energy-momentum "tensor" of matter and gravitation" is "locally conserved."

p. 155: "The term [lambda]g_mv was originally introduced by Einstein for cosmological reasons (which have since disappeared); for this reason, [lambda] is called the cosmological constant. This term satisfies the requirements (A), (C), and (D), but does not satisfy (E), so [lambda] must be very small so as not to interfere with the successes of Newton's theory of gravitation. Except in Chapter 16 (cf. "the scale factor R(t)", p. 613 - D.C.), I am assuming throughout this book that [lambda] = 0."

Comment (D.C.): After the discovery of the "dark" energy in the fall of 1997, Steven Weinberg didn't re-examine the source of the problem: you get "dark" energy because you presume that it comes from stuff with positive energy density, but if you focus on the puzzle of the physical existence of one "charge" only, you may discover the mechanism by which we see only positive energy densities, and not 'torsion connection' in GR. It should be "dark", of course.

Regarding the dynamics of "the scale factor R(t)", and the meaning of the variable t therein, see my email and Fig. 2 above.

May I ask two questions. You wrote, in your latest arXiv:1006.2154v1 [gr-qc], that torsion energy has "a pure geometric origin", and "follows a
conservation law, similar to that of the curvature energy (for details see reference [7])".

Would you consider the other "charge" of mass, along the lines of Hermann Bondi (Negative Mass in General Relativity, Reviews of Modern Physics 29 (1957) 423)? Can you suggest a mechanism by which we can observe only positive mass, hence the physical source of torsion energy is not traceable in GR, and its origin looks like "a pure geometric origin"?

Brian P. Dolan (September 21, 2010), Lecture Notes MP476: Cosmology, Sec. 3.1 Cosmological Constant, p. 45: "In General Relativity R is not a radial co-ordinate, it is a length scale determining the physical size of lengths in 3-dimensional space. If R = R(t) then when ˙R > 0 space is said to be expanding while when ˙R < 0 space is contracting. We can interpret R as the physical distance between any two fixed galaxies, provided their separation is of the order of 100M Pc, ormore."http://www.thphys.may.ie/Notes/MP467/Cosmology-Lectures.pdf

Dear Brian,

Regarding the quote from your Lecture Notes, please recall that if something walks like a duck and quacks like a duck, it may not be necessarily a duck ("the physical distance between any two fixed galaxies"). Details in Sec. Summary, pp. 35-36, at

2.1. "The cosmological principle supposes that the Universe is spatially isotropic and homogeneous. In particular, this implies that there exists a privileged class of observers, called fundamental observers, who all see an isotropic universe around them. It implies the existence of a cosmic time and states that all the properties of the universe are the same everywhere at the same cosmic time. It is supposed to hold for the smooth-out structure of the Universe on large scales.

"We can distinguish it from the Copernican principle which merely states that we do not live in a special place (the center) of the Universe. As long as isotropy around the observer holds, the principle actually leads to the same conclusion than the cosmological principle."

2.1. Comment: Regarding 'fundamental observers', see Eq. 1 on p. 35 here. In the words of Paul Valery, "God made everything out of nothing, but the nothingness shows through."

As to the Copernican principle, recall the old wisdom that the universe looks like an unbroken ring with no circumference, for the circumference is nowhere, and the "center" is everywhere. The topology of such universe is unknown, of course.

2.2. Regarding Sec. 1.1.7, 'The equation of state of dark energy'

Jean-Philippe Uzan: "The equation of state of the dark energy is obtained from the expansion history, assuming the standard Friedmann equation."

2.2. Comment: Perhaps we could rely on the "expansion history" if only we knew the variable denoted by t in R = R(t) from Brian Dolan above. The problems are enormous.

2.3. Jean-Philippe Uzan, p. 41: "At the moment, none of these three possibilities is satisfactory, mainly because it forces us to speculate on scales much beyond those of the observable universe."

2.3. Comment: See the answer to the crucial question above. If you employ Archimedean geometry only, there is nothing at your disposal to define ds & dt and 'the grin of the cat without the cat' (Alice).

3.1. Regarding Sec. 7.2 (pp. 104-105), Jean-Philippe Uzan wrote (p. 103): "The numerical values of the fundamental constants are not determined by the laws of nature in which they appear."

3.1. Comment: In the same vein, the source of "dynamic dark energy" cannot be located inside the very thing it produces: 3-D space. I tried to explain the puzzle to my teenage daughter here; details and implications here.

> Weinberg does not mention it for a very good reason ---> there is no known source of matter that produces any> appreciable torsion that could affect the Friedmann> equation and influence gravitational dynamics, it is> simply too weak. That is why it usually ignored.

I believe have discussed, at the link sent in my initial email, the problem with the so-called "dark" energy. Its origin may not be confined exclusively to some (i) matter source with (ii) positive energy density (as usually done in the old cosmological problem by addressing the question of whether the quantum vacuum "gravitates", by Wolfgang Pauli in 1930s).

The "dark" energy may be of geometrical origin; hence my interest in the origin of torsion effects.

If we zoom on the affine structure (cf. Graham Nerlich below), I believe one can speculate about two "dark" components (global mode of spacetime), which I call 'torsion' and 'curvature' components of the affine connection. Please note that such two-component affine connection is postulated to explain how some Hausdorff topological space gets "connected" and can be endowed with differentiable structure.

In other words, all this is postulated prior to GR textbooks and review articles on torsion (e.g., how tangent spaces twist about a curve when they are "parallel transported"). It's all about the purely geometric connection "between" (=global mode of spacetime) points; please see Fig. 2 at

It's a whole new ball game "between" the points of the manifold: the curvature component of the affine connection pertains to the "bridge" between two adjacent points on a 'line', while the torsion component of the affine connection refers to this same "bridge" between two adjacent points, made by a 'cycle' (cf. above). It's the topology of the "bridge" that makes the difference between the two components of the affine connection.

In some ludicrously fictional "flat" (Minkowski) space, these two components of the affine connection should be "flattened", hence made indistinguishable, which would in turn obliterate the affine connection itself.

"General relativity set the affine torsion to zero, because it did not appear necessary to provide a model of gravitation (with a consistent set of equations that led to a well-defined initial value problem)."

Do you believe that GR can eventually produce *any* 'well-defined initial value problem' ? If you do, please show me one example of solved Cauchy problem for the Einstein equations.

All the best,

Dimi------

Note: Let me quote from an alternative viewpoint [Ref. 1], in which the KS Theorem and Free Will Theorem haven't been mentioned. Notice that your brain has no choice but to follow its states, which are fixed in the "past, present and future all at once." All this explained with lots of advanced math. Enjoy!

Footnote 1, p. 1: "S. Lloyd tells the amusing anecdote[?], “I recently went to the National Institute of Standards and Technology in Boulder. I said something like, ‘Your clocks measure time very accurately.’ They told me, ‘Our clocks do not measure time.’ I thought, Wow, that’s very humble of these guys. But they said, ‘No, time is defined to be what our clocks measure.’

"Indeed, the standard second is defined as the duration of 9,192,631,770 oscillations of the radiation from the transition between the two hyperfine levels of the ground state of the cesium 133 atom.......

p. 30: "In general relativity, for example, we have an initial value formulation, but can also find global solutions. In the initial value formulation, we can specify the configuration of the world at a given time, then integrate forward to predict how things will evolve. However, in the case of a global solution such as a cosmological model, we are presented a complete description of past, present and future all at once. In this view, the outcomes of measurements are already fixed.

"The best we can do is to think of consciousness as sequentially illuminating certain fixed events, then others, with all the events already right there in the solution."

On Sat, 1 May 2010 14:00:41 +0100 (IST), you wrote:>> On Thu, 29 Apr 2010, Dimi Chakalov wrote:>>>> The "dark" energy may be of geometrical origin; hence my interest in>> the origin of torsion effects.>> I guess this comes down to the old chicken and egg problem on Einstein's> equations, which Einstein himself was not happy about.> The geometry is determined by the matter distribution, but you> need the geometry to specify the matter distribution. This is> just as true in the Einstein-Cartan formulation as in the> original Einstein formulation.

I think the chicken-and-egg problem and the so-called Buridan donkey paradox

encapsulate the *paradox* of time in GR, as a relational/background-free theory: "the metric is treated as a field which not only affects, but also is (at the same time - D.) affected by, the other fields" (John Baez). This paradox is depicted by Escher hands,

Hence the dynamics of GR cannot be resolved with the unsolved puzzle of 'time in GR'. It is not surprising to me that people cannot define some 'gravitational energy' that would be conserved in such paradoxical time, and in the case of the Einstein equations alone, "there are no physically motivated boundary conditions" (Alan Rendall). Which brings me to the discussion with Laszlo regarding the precise meaning of his statement that any observable of the gravitational field is "necessarily quasi-local",

> If so that should translate into a statement about matter which,> assuming it has observational consequences, could be tested.> Historically I think people have worked the other way --- they look for> types of matter that might give torsion and all the known> candidates seem to give negligible torsion. If you have> a geometrical principle in mind that makes torsion significant> that would surely have observational consequences?

My "theory" is assembled on most general level. I cannot derive quantitative predictions, so it's far too early to say anything conclusive about the "dark" geometric effects of torsion.

Moreover, please bear in mind that everything I've written on my web site could be wrong. After all, I'm just a psychologist,

Dimi------------Note: Regarding "a statement about matter which, assuming it has observational consequences, could be tested": use your brain to test 'potential reality' here, and check out the UNdecidable quantum state here. It's not about type I matter fields. It's all about quantum gravity.

Pity Brian Dolan didn't have time to check out the links. Your wristwatch does read the standard second defined as "the duration of 9,192,631,770 oscillations of the radiation from the transition between the two hyperfine levels of the ground state of the cesium 133 atom" (reference above). If you apply here GR, "points become fuzzy and locality looses any precise meaning", says Sergio Doplicher. Hence your wristwatch reads a finite time interval, called 'second', based on the miraculously precise timing of the cesium 133 atom. Moreover, the fact that "it is possible to look around, and see as far as we can" (Lee Smolin) is another mystery that GR cannot explain either.

The mystery of these finite intervals of time and space, comprised of infinitely many and infinitely small "points", is the subject of the so-called Relative
Scale Principle (RSP) outlined here. Central to
RSP is the hypothetical pre-geometric plenum, which is supposed to act as a connecting-and-separating object that does not belong to this perfectly smooth set of "points". Hence we may describe an emergent and perfect continuum of "points", called 'local mode of spacetime': please see Fig. 1 above. That's 'law and order' at all length scales, from the Planck scale to the sliding cosmological "horizon". Pity nobody is interested.

Final version is expected on 25 November 2015,commemorating the 100th anniversary of GR(joint work by Levi-Civita, Einstein, and Hilbert)

Abstract

It is argued that the Hamiltonian formulation of GR cannot address the dynamics of space due to its "dark" energy from the quantum vacuum, hence new degrees of freedom should be introduced to the dynamics of space -- an arrow of space. The scope of 'relative
scale principle' (RSP), announced on 21 September 2008, is to introduce "boundaries" on spacetime, such that an isolated system endowed with 'finite infinity' can be constructed. The ontological assumptions in RSP are about 'necessary and sufficient conditions for spacetime': the former concerns physical substratum (positive energy density), while the latter condition refers to a global, Heraclitean, and non-Archimedean state of the whole universe as ONE.-------

[Excerpts from the 3rd draft version, 2 August 2010]

Let me start with a quiz: Look at the two pictures below and tell the essential difference.

Yes, you got it right: the first picture is a negative image from the original.

You may also say that, unlike Chuck Norris, those people at NASA, who offer a meta-observer view on the "expanding" 3-D space, use lots of math, but that doesn't really matter, because the essential similarity is that both images are jokes. Back in 1963, Roger Penrose offered a recipe for reaching infinity from any location in space, which was also a joke, although spiced with lots of math.............

Here's the problem of "more space". Some prelims from GR textbooks: unlike STR, the spacetime in GR textbooks is considered "dynamical", such that, to follow a well-know metaphor, the "actors" (matter) and the "stage" (spacetime) engage in a non-linear negotiation (Escher's hands): "Space acts on matter, telling it how to move. In turn, matter reacts back on space, telling it how to curve" (John Wheeler). But notice that, unlike GR textbooks, the non-linear negotiation between the two sides of Einstein equation is interpreted here as "taking place" in the global mode of spacetime endowed with additional spatial degrees of freedom from the arrow of space.

We all agree that 'time' is no longer a fixed background parameter, but the global mode of time, in which the non-linear negotiation of geometry of spacetime vs. matter "takes place", cannot be read by any wristwatch from GR textbooks. This produces a plethora of "times" in GR textbooks, but "these definitions of time are in general unphysical, in that they provide no hint as to how their time might be measured or registered" (Butterfield and Isham). Here, all these "times" constitute the "eternally present" all time [tau] (Karel Kuchar)..............

To elucidate the second major difference in treating the 'tangent vectors', recall that in GR textbooks you are invited to take the same meta-observer, bird's-eye view on the whole spacetime as in the NASA picture above: to explain the alleged "curvature" of spacetime, "it is easy to see it in a 2-dimensional surface, like a sphere. The sphere fits nicely in 3-dimensional flat Euclidean space, so we can visualize vectors on the sphere as 'tangent vectors'" (John Baez). Then you've been taught by Bob Wald "to work infinitesimally, using the idea that, on sufficiently small scales (this notion certainly needs clarification in the context of Thompson lamp - D.C.), a curved geometry looks very nearly flat (notice the poetry - D.C.). These departures from flatness can then be described via differential calculus. To do so, one begins by introducing the notion of a tangent vector to describe an infinitesimal displacementabout a point p ." The major differences to the interpretation offered here are in the following. Regarding John Baez, the local mode of spacetime is not embedded in any "higher-dimensional flat spacetime", but in an infinite-dimensional purely geometric manifold (called here 'global mode of spacetime'). Hence 'the infinitesimal' is indeed equipped with 'tangent vectors', but they all "point" to the global mode of spacetime -- not to the local mode.

The new "direction" from the "dark" energy is non-existent in ADM presentation, since it pertains to the 'absolute time' of the 'absolute structures'. Small wonder Karel Kuchar couldn't dig it out from present-day GR: you have a plethora of unphysical "times" from the foliation recipe, and no reference fluid (global mode of spacetime), which makes you 'eternity blind' (John G. Bennett).

The crucial "direction" of the so-called "expansion" of space due to the "dark" energy from empty space (Lawrence Krauss) requires a brand new dynamics of GR. In GR textbooks, the flexibility of space is limited to its "curvature" only, and, if you manage to compute a linearized snapshot from Einstein's filed equation, you inevitably end up with a frozen "block" spacetime: "There is no dynamics within space-time itself: nothing ever moves therein; nothing happens; nothing changes" (Robert Geroch). Then the new dynamics of spacetime, due to the "dark" energy, can only spring from something "outside" (the local mode of) spacetime. That is, from "outside" both sides in the Einstein equation -- the global mode of spacetime. It literally creates both "more space" (cf. R. Rakhi & K. Indulekha and S. Carroll above) and "more matter".The proper dynamics of 'GR with DDE' cannot be unitary. Enter the puzzle of "points": the perpetually re-created world of facts (local mode of spacetime) and its ultimate source 'the ideal monad without windows' (Döring and Isham) contain uncountably infinite points ('the set of all sets', maybe). The cardinality of such "set" (if any) is not even aleph-0. Once created by The Beginning, the local mode of spacetime is wrapped by its special "boundaries" (see RSP below), such that the 3-D space of the local mode is de facto infinite (dual age cosmology)............

On the other hand, the 3-D Flatlanders (local mode of spacetime) will only notice that the two 'ideal endpoints' or "edges" of the local mode of spacetime (denoted with S and L, see below) have acquired some 'numerically finite but physically unattainable values' (one of which is the Planck scale), and will recall the old idea that the universe looks like an unbroken ring with no circumference, for the circumference is nowhere, and the "center" is everywhere.

But they will also notice some puzzling "projection" from the dark energy, cast on the 3-D space, just as they can notice, and indeed measure, the "projection" of space curvature, cast on the 3-D space (see Larry Krauss, p. 12). While we don't know the nature of gravity, we are accustomed from GR textbooks to treat it as 'curvature of spacetime', although it shows up more like 'deviation of something from something else that might be "flat" but you can't tell, because you have only one "flat" point'.

The "projection" from "dark" energy, cast on 3-D space, is just as weird and misleading as the 'curvature of space', yet it bothers people much more than the puzzling nature of gravity (the latter has to disappear on "points", so you have only one "flat" point as a reference object for "curvature").

People from NASA may ask, what kind of stuff is "expanding", creating the illusion (see also the misleading picture below) about galaxies "running away" from each other, in line with the Hubble Law?

Wrong question. In the local mode of spacetime, nothing could possibly "expand" or be "curved". The actual illusion is that something is "expanding" (or "curved"). It is the illusion from NASA's meta-observer view.

NB: All such illusions, the absolute structure of space included, should be eliminated by proper relativistic presentations, such that the sole invariant object in space will be an infinitesimal "point". Its 3-D projection or "shadow" does look like a "point", but upon a closer view it unveils its infinite-dimensional nature.

The picture below (as well as the one from NASA above) can only be seen by some meta-observer, who can also see the whole universe, and of course count to infinity (like Chuck Norris). The 3-D Flatlanders can't.

Such kind of illusionary "expansion" (much like the "curvature" of space) may be caused by some 3-D "projection" from something, maybe some process or mechanism, which takes place in the global mode of spacetime. How about the old idea of the mutual penetration of the Large and the Small?

Perhaps Einstein would only look at my efforts and say ... well, he was a very polite person. But because he also was, on some occasions, driven solely by his personal taste and instinct, rather than sound physical principles [Note 1], let me try to follow his style.On 21 September 2008, I suggested a scale relativity principle (SRP) aimed at explaining the nature of space (don't confuse it with Laurent Nottale's scale relativity). If we are to think as proper relativists, I believe we should abandon the absolute structure of 3-D space in today's GR [Note 2], and work out new symmetry presentations of The Large and The Small.

I also admit that SRP is direct prediction of my theory, hence if it turns out to wrong, so will be my whole project, as started in January 1972.

I believe physical objects can be considered "large" bzw. "small" only with respect to the macroscopic length scale of tables and chairs. If a volume of space changes its size toward 'the Large/Small', the metric of space in 'the Large/Small' will be altered accordingly, in such way that the "size" of objects in that sphere or volume will remain unchanged in the volume-determined reference frames of 'the Large/Small', yet will continue to look 'Large/Small' in the initial macroscopic scale frame of classical mechanics. The idea of SRP involves (i) an omnipresent observer (Claudia) endowed with 'the eternally present all time [tau]' (Karel Kuchar), (ii) space inversion producing CPT-invariant presentations of 'volume of space', and (iii) macroscopic observer located at the length scale of classical mechanics, from which the postulated mutual penetration of the Large and the Small starts to unfold.

We have three possible views on 'the length of objects' in 3-D space. First, the viewpoint of a macroscopic observer who can simultaneously see (i) 'the Large' as an expanding volume of space, and herself being located inside such expanding sphere, and (ii) 'the Small' as contracting volume of space, and herself being located outside such contracting sphere.

Consider a table with length two meters, located inside a sphere with diameter two meters, along with the macroscopic observer inside this sphere. If the table is stretching toward the size of a galaxy, it will look like 'getting larger' to the first observer, but its intrinsic length will always remain "two meters" to a co-moving observer in its scale-dependent reference frame evolving toward 'the Large'. The co-moving observer will never, in no circumstances, register any change of the length of the table, as her metric also expands with the same rate as the expanding table toward 'the Large'. Likewise for the journey toward 'the Small': take the first observer and the two-meter sphere at macro-scale, perform space inversion, and now the CPTmirror image (like turning a rubber glove inside-out) of the macro-observer will find out that she is now located outside a contracting sphere, as the table is now running toward the Planck scale, yet a co-moving observer will again see "her" table as keeping its intrinsic length of 'two meters' in her scale-dependent reference frame toward 'the Small'. This is the 'second view'.

And the third view is that of a fixed meta-observer (global mode of spacetime), who will see a table stretched to the size of a galaxy, a table at the initial macroscopic length scale, and a table squeezed to the size of an atom, all of which possessing 'the same UNdecidable (KS-like) length'.Now, one can offer an operational definition of 'infinitesimal volume of 3-D space' in the local mode of spacetime (in ADM presentation, cf. B. Bolen, it is denoted with ds). Imagine an omnipresent observer being "spread" inside an 3-D Euclidean sphere -- except for the center. She looks at the point-like center of the sphere from infinitely many directions (radii) -- simultaneously.

But the center of the sphere is not a bona fide mathematical point -- it is an 'infinitesimal', such that one can treat it as 'the smallest possible rubber glove' and perform CPTspace inversion, by replacing (not 'reversing', as in Wu-Ki Tung's textbook, Ch. 11) all "points" inside 'the infinitesimal rubber glove' with all points outside it, that is, with all points of Claudia. Notice that we cannot use quantitative notions like 'more' and 'less' regarding any "number" of points -- neither those of Claudia nor those inside 'the smallest possible rubber glove'. Were the 'infinitesimal' an abstract mathematical point, the operation 'space inversion/replacement' would be impossible, and then there would be two opposite and privileged views on 'the Small' and 'the Large', and the latter would be 'absolute qualities of space', just like in GR textbooks.

Thus, an operational definition of 'infinitesimal volume of 3-D space' is that which permits space inversion at (i) the minimal, hence infinitesimal, volume of space (S) and (ii) an arbitrarily large (inverse-proportional to S) volume of space (L).

Now an omnipresent observer can simultaneously "see" the very same center of the sphere (infinitesimal volume of 3-D space, S) in a space turned "inside out" as well; one could perhaps say that she is "residing" at two superposed inverted states of 3-D space, as she is able to see both S from L and L from S -- simultaneously. She is not restricted (as we are) to choose one of the 'views'. She is placed at S = L, and is wrapping the whole (seemingly "expanding", according to the first macroscopic observer above) 3-D space en bloc.............

It doesn't matter whether this shrinking or expanding is (or is not) "accelerated": either way, the "number" of points is a non-Archimedean notion that cannot be subject to quantitative evaluations like 'more' vs. 'less'.

What matters is that the good old 3-D space has just been re-created, like a Phoenix Universe, along the arrow of space. The mechanism is the same as in The Beginning, but without the initial "inflationary stage" -- just an elementary step of the arrow of spacetime and an elementary increment of the memory of 'the universe as a brain', which leads to "more points" being emerged in its memory, available to choose from in the next elementary step: the universe only gets smarter, richer in terms of physical content, and more flexible, just like an evolving brain. If some day [John 1:1] decides to halt the arrow of spacetime, the UNdecidable nature of 'the infinitesimal' will be nullified, the Thompson's lamp paradox will suddenly become solvable, the number of "points" will become denumerable (e.g., there will be roughly 1099 "atoms of volume" in every cubic centimeter of space, as claimed by L. Smolin, p. 61), and the 3-D space will cease to exist. From our viewpoint, such development is logically impossible (the vacuum cleaner paradox)............

The global mode of time runs simultaneously in opposite directions -- in all possible directions in 3-D space -- which will make her (and ours) unanimated wristwatch dead frozen, like the proper time of a photon "between" emission-and-absorption; see Kevin Brown. (This is the exact meaning of 'Your Global Time is ZERO', which was posted on this web site on March 14, 2004, to commemorate Einstein's 125th birthday; see the atemporal "handshaking" medium above.)

Such symmetry over 'space inversion' and the two "superposed" inverted views of Claudia (needed for derivation of three views on 'the length of objects' in 3-D space; see above) reveal the sole invariant object in 3-D space -- the infinitesimal volume of 3-D space, called "point"............

After this shaky exercise with the sole invariant object in space, called "point", we need to explain the two 'scale factors' (for the lack of better terms), which create 3-D space: S (from Small) and L (from Large). S refers to 'the infinitesimal point', and L to 'the maximal yet finite volume of an ever-expanding 3-D space'.

We postulate: LS = 1 . This is the recipe for a natural "boundary" (cf. G F R Ellis' Finite Infinity) imposed on L due to the "extension" of the infinitesimal "point" S .

If we think of The Beginning as the case in which L equals exactlyS (L = S = 1), we can understand the "inflationary stage" in which a tiny little difference between L and S is being introduced by [John 1:1], after which the two scale factors will run in opposite directions, to produce 3-D space tightly filled -- perfect continuum -- with infinitesimals S and bounded by an "expanding diameter" L . Now Claudia will be placed in the global mode of spacetime -- both inside S and outside L . She will be able to "see" the whole (infinitely-connected, by Claudia) local mode of spacetime en bloc, and her global mode of time will be just like that of our brains: an atemporal and infinite-dimensional "handshaking" medium by which each and every infinitesimal S is simultaneously correlated -- the 'absolute now' of Claudia -- with 'the rest of the universe', as depicted in the Escher drawing below. And since their EPR-like correlation is "spanned over" an infinitesimal "point", in the local mode we will see the blueprint from quantum waves and relic inflationary waves, yet their source will be inevitably "dark". (In order to detect quantum waves "online", one would have to perform a perfect non-demolition measurement avoiding any point-like collapse, by screening the quasi-local wave over extended 3-D domains.)

Notice that Escher's hands can be moved only by an arrow of space, not with the kinematics of L and S alone.

Briefly, we can think of The Beginning as some "absolute vacuum polarization" (not necessarily irreversible), and suggest an equation of space:

L = S = 1 (John 1:1 )--> LS = 1

The equation above contains reference to The Gospel, but I believe St. John wouldn't mind to be removed by those obsessed with different religions, such as anti-theism or "agnosticism" (those who deeply believe, for the lack of scientific proof, that there is nothing to believe in).

In summary, the elementary building block of geometry, presented with a geometrical "point" (explanation and drawings here), is supposed to be 'at the same time' a space-inverted image of 'the whole universe' at L , and vice versa. An omnipresent observer in the global mode of spacetime (the two "superposed copies" of Claudia) will be able to monitor all the "points" in the whole 3-D space en bloc, since it is ONE single object, L = S . Our "Claudia" is nothing but the postulated Aristotelian Connection, which binds all "points" by/via their space-inverted image at the level of ONE (the whole universe at L ).

This is the pre-geometric plenum, which is still missing in differential geometry textbooks. Consider, for example, Chris Isham's Modern Differential Geometry for Physicists, 2nd ed., in which you are advised that the Hausdorff topological space has been somehow "connected" (p. 61, footnote 1; see also Sec. 1.2.1, p. 3, and pp. 59-60 and Fig. 2.1 therein). It is unclear what sort of pre-geometric plenum could do this job. Think of it as 'the universe as ONE', which is being multiplied as infinitely many (uncountably infinite) geometrical "points" ensuing from 'the ideal monad without windows' (Döring and Isham), in such way that all "points" are connected and correlated by/via 'the universe as ONE' at L = S .

This is a very old idea 'all is in all', which I am trying to cast in some (relatively) comprehensible relative
scale principle (RSP). There are no "genuine" nor "absolute" dimensions of physical bodies, according to SRP. The spatial relations of 3-D space -- inside/outside and left/right -- should originate from a four-segment "lily" spacetime diagram of the global mode of spacetime (notice the fav icon of this web site). Once such four-segment spacetime diagram is properly constructed (we also have to explain rotation symmetry), we should be able to calculate the cancellation of 'all but one' potential states of quantum-gravitational systems, hence show the correlated states that build up the local mode of spacetime, valid for only one frozen snapshot from Die Bahn.

The frozen snapshot from the "film reel" (the arrow of spacetime) will display an already-correlated set of point-like values of all physical variables, in line with the Bootstrap Principle 'think globally, act locally'. The atemporal (with regard to our unanimated wristwatches) correlation "takes place" in Claudia's global mode of spacetime. She is residing in (L = S), and her two "superposed copies", toward L and toward S, contain uncountably infinite "number" of classical-able states from which one correlated and re-created physical universe is being explicated as 'local mode of spacetime': one-at-a-time, along the arrow of spacetime.

Notice, however, that the direction of the arrow of spacetime points only and exclusively only to the global mode of spacetime, but the latter is totally hidden in the local mode: the "dark gaps" are completely sealed off by 'the speed of light', rendering the local mode a perfect continuum.

Another important feature of SRP is the difference between L and S in their postulated relation LS = 1 : while S is "bounded from below" by a geometric "point", the ever-expanding "horizon" of space is bounded by an entirely different geometric object, which is (sort of) 'inverse-proportional' to a geometric "point". Although the "number" of points needed for L and S to 'pass through' in order to reach the exact relation LS = 1 is indecisive in the local mode of spacetime, there is nevertheless a difference between the "run" toward S and the "run" toward L , which determines the asymmetry of time. After all, in the global mode of spacetime the age of the universe is finite, such that any "run" towards it will have a fixed cut off at The Beginning, while the "end" of the cosmological time is indecisive, which in turn permits the evolution of 'the universe as a brain'.

NB: Notice the nature of the cut off toward S : it cannot be reached from within the local mode of spacetime due to the inherent teleological structure of 'the world of facts'. This truly fundamental cut off can only be "seen" from the global mode of 'the whole universe as ONE', hence it is considered to be the Aristotelian First Cause. Its action is called The Aristotelian Connection. It acts without being 'acted upon'. Its job is to cast point-like values of physical quantities at the level of 'geometry' -- the grin of the cat without the cat, as Alice would have said regarding the left-hand side of Einstein equation.

And because The Aristotelian Connection produces not just one "point" (as in the Thompson Lamp paradox) but an infinite "number" (uncountably infinite) of such purely geometrical "points" -- simultaneously, with zero duration according to your wristwatch -- it connects these "points" of the local mode of spacetime as a pre-geometric plenum. It is also the ultimate 'chooser' of particular physical content that would belong to particular geometrical "point": due to the correlation and negotiation (cf. Escher) in the global mode of spacetime of all potential physical content available to the local mode, the 'chooser' of the particular physical content specifying any particular "point" is 'everything else in the universe'. The end result is the cancellation of 'all but one' state from the spectrum of potential states available to the particular "point" to choose from, and we get the local mode of already-correlated facts -- one-at-a-time, along Die Bahn (the arrow of space).

Nobody and nothing "plays dice" here: God casts the die, not the dice. The inherent flexibility (not "uncertainty") of all spacetime points to get dressed with specific physical content is like the flexibility of the human arm to perform any movement. Surely there are constraints, boundary conditions,
conservation laws, etc., yet the flexibility of all "points" to choose from their quantum-gravitational spectrum of potential states (God's "die" or rather 'matrix') can never be reduced to zero. For example, Heisenberg's relations can be interpreted as the flexibility of a quantum particle to choose from a spectrum of potential states: if its potential states regarding its next 'position' get shrunk, it will have a corresponding largerspectrum ofpotential states regarding its next 'momentum' to choose from. Of course, if you "filter" this quantum flexibility through the spacetime of STR, you may claim that it is impossible for a quantum particle to possess simultaneously point-like values of "complementary" observables. But again, these values need not be point-like. They can be perfectly well-defined yet quasi-local. The idea of 'wave-particle dualism' may be correct for QM measurements, but nobody knows, for example, how would an electron look like in an intact quantum world: if it is not forced to behave like a point-like particle since it isn't being "observed" at the length scale of tables and chairs, how would its "wave" look like? Perhaps the quantum "wave" of an intact electron pertains to its quantum flexibility.

Going back to the pre-geometric plenum: perhaps the asymmetry of time is determined by the asymmetry of space, namely, by the difference between the two "runs" toward L and S , bounded by their relation LS = 1 . All efforts to derive the fundamental asymmetry of time from some physical stuff or 'entropy' are doomed to fail, because such fundamental feature of spacetime can only be determined by 'the grin of the cat without the cat'.

Also, our wristwatches [Note 3] are perfect examples for quantum-gravitational measuring devices, as they can in fact read one 'dynamical instant' from Claudia's absolute time since The Beginning, but because we can "look" at this dynamical instant only with a physical "torch", we inevitably see a dead frozen snapshot, as explained eloquently by Robert Geroch.

Everything said here pertains only to the kinematics of spacetime; to obtain its dynamics (the arrow of spacetime) we need to include the Aristotelian First Cause "by hand", because nothing in this purely kinematical structure can or should point directly to God.

In modern parlance, SRP is expected to "suffer" from Gödel's incompleteness theorem, but because of the fundamental nature of SRP, the additional elements in it, which belong to some 'meta theory', will inevitably come from 'outside science'. To be specific, the introduction of Aristotelian First Cause "by hand" means endowing the universe with the faculty of self-acting: just like the human brain (we think about our brain, with our brain), the universe modeled as a 'brain' should have the faculty of acting on itself, but no physical path toward 'the acting mind of God' (such path may show up with math only) should show up in quantum gravity. Perhaps some of those "agnostic" people may prefer to call Him "dark" -- I don't mind; we all enjoy His gift called free will.More in my talk in November 2015. Just a hint: look at Fig. 2 in the note about Die Bahnabove. The direction of w is "dark" to the local mode, and the gravitational effects (global mode) should have 3-D presentations (e.g., the so-called "curvature") in the local mode (Fig. 1). If some day we find out that the origin and the mechanism of inertial reaction "forces" are indeed produced by the gravitational potential reality, perhaps we will explain the most difficult puzzle: the asymptotically flat space of the local mode of spacetime, produced by the two tug-of-war components of gravity at cosmological scale: CDM and DDE.

Perhaps at the largest length scale, L , the "dark" geometrical effect from the global mode makes the 3-D space of the local mode 'asymptotically flat'. However, given the basis relation LS = 1 , an absolute flat and absolute infinite 3-D space would require some mathematically unclear state {S = 0 , L = [inf]} : the two symmetry axes of the infinitely-inflated sphere and torus will coincide, along with their centers (check out a story about a screen saver in Windows 98 here). I intend to argue, in November 2015, that the topology of the local mode of spacetime is being dynamically fixed as 'asymptotically flat' at each and every step of re-creation of the local mode (cf. Die Bahn above), by "passing" through such mathematically unclear state that is neither sphere not torus anymore. Stated differently, I suppose there exist two asymptotical "boundaries" on what looks like an 'asymptotically flat 3-D space': a 3-D sphere and a 3-D torus with the largest possible radii, L , "between" which a mathematically unclear (and purely theological) state is quietly residing. All this will be a sequence to my previous talk in September 2008.

As to the "expansion" of space depicted in the drawing from NASA, perhaps the projection of the metric in 'the Large' onto the macroscopic scale produces the illusion to the 'first observer' (cf. above) of some "expanding universe". Again, according to
RSP, the "expansion" is not caused by any physical stuff. It is an illusion of some "expanding" metric, which can be seen only by the 'first observer'.

As suggested above, the cosmological horizon is 'ever expanding' along the arrow of spacetime, while the Planck scale is fixed. That is, the 'infinitesimal' is fixed (bounded from below) by some 'numerically finite but physically unattainable' values that do not change (e.g., the Planck length), while the 'maximal volume of space for gravity' (L) increases with the age of the universe, like an expanding horizon "bounded" by an ever-sliding L : Panta rei conditio sine qua non est.By the way, have you seen an UFO with your very eyes? If these guys can switch off the inertial reaction "forces", perhaps they can fly effortlessly, much like we move our thoughts. And if they can modulate the metric of space, perhaps they can make their pathway in the whole Milky Way just "two meters". From their perspective, they will indeed fly 'two meters', while the space will be "running towards them". Perhaps in such altered-metric space they fly literally 'from point to point', and with a very low speed of, say, just 2 m/s, as recorded by their wristwatch.

Wilbur B. Smith: "... to produce the gravity differential, the time field differentials which were necessary to operate the ship."

I wish I knew what is 'space' [Note 4]. There is so much latent energy packed in the quantum vacuum. Perhaps all we may need is access to the global mode of spacetime. The whole
RSP may sound unclear and farfetched, but recall Christopher Columbus: If we don't leave for India, how can we discover America?

D. ChakalovFebruary 18, 2010Latest update: August 2, 2010

Note 1. In a letter to Paul Ehrenfest from 4 February 1917, Einstein wrote about his 'cosmological constant':

"I have again perpetrated something relating to the theory of gravitation that might endanger me of being committed to a madhouse." (Ich habe wieder etwas verbrochen in der Gravitationstheorie, was mich ein wenig in Gefahr bringt, in ein Tollhaus interniert zu werden; English translation by N. Straumann).

"The right side (the matter part) is a formal condensation of all things whose comprehension in the sense of a field theory is still problematic. Not for a moment, of course, did I doubt that this formulation was merely a makeshift in order to give the general principle of relativity a preliminary closed expression. For it was essentially not anything more than a theory of the gravitational field, which was somewhat artificially isolated from a total field of as yet unknown structure."

Note 3. The problems with 'time in GR', the paradox of time in canonical quantum gravity (Karel Kuchar), and the puzzle of so-called "dark" energy from empty space stem from the absence of the arrow of spacetime in present-day GR. On the one hand, our mischievous wristwatches do in fact read the arrow of spacetime as a "chain of points", as they "filter" only one of the infinitely many "dynamical points" from the global mode of time, and read these filtered points "one-at-a-time" along the "vertical skewer" of the arrow of spacetime (global mode of time), on which all "horizontal snapshots" are stacked. On the other hand, in order to "look" at this arrow with physical (unanimated) devices, we can only use a "torch", which kills the arrow, and shows only onefrozen "point" fixed by the calculated values of physical quantities at this dead frozen "point". And here people like Robert Geroch say -- let's try to solve the Einstein field equations, starting from this well-calculated frozen "point", and then move the (canonical) data only on this "horizontal" set of such "points", and end up with the insurmountable Cauchy problem for the Einstein equations. Why? Because the genuine dynamics of GR ain't there (Stanley Deser will undoubtedly disagree).

Note 4."In the first place, we entirely shun the vague word "space," of which, we must honestly acknowledge, we cannot form the slightest conception."

---------P.S. Nobody has so far asked, what is the meaning of the phrase 'Your Global Time is ZERO'. The sole question from the readers of this web site has been, 'why don't you publish all this in a book, to explain your ideas' (or something similar).

The criticism is fully justified: reading my web site is anything but fun. I am considering writing a book, but it will be intended to kids age 15+, and will have to be complemented by a
DVD with video lectures, to explain the crux of the idea about 'the universe modeled as a
brain'.

Why kids? Because the future belongs to them (I will soon hit 58, Deo volente). My work on quantum gravity concerns very few people, all of whom have this utterly negative attitude of 'the worst of criticism is neglect'. Just a few examples: the last time I heard from Claus Kiefer was in 2003, only to tell me that he can't open the CD ROM I sent him by surface mail, because all computers at the University of Cologne run on Unix. The last time I heard from Karel Kuchar was on 8 Jan 2003, and from John Baez on 14 Jan 2002. The case with Chris Isham was even worse. And look what happened with my talk in 2008.

Thinking about the future, I can imagine only two developments regarding
RSP. One possible case will be that it is just crap and delusion, so obviously wrong that none of the renowned experts made the slightest effort to show my stupid dilettante errors. Will see. Planck's Law of Thermal Radiation was met with very nasty attitude by the established scientific community [Note 5], as it literally blew away their comfortable world. If
RSP turns out to be correct, it will cause far more "damage" to many established scholars studying canonical quantum gravity and "gravitational wave astronomy" (at least 679; see also [Note 6]).

But could
RSP be correct, really? Well, as I mentioned previously, the two rules for success are:

And nobody is curios anyway. (For those who are: I believe RSP, embedded in an arrow of spacetime, offer the only possible solution to the mind-brain problem -- the mind has to be both detached from matter, to preserve its ontologically different nature, and linked to it, in order to communicate with its brain via the bi-directional "talk" depicted in the Escher drawing above.)

Therefore, there is no need to use paper -- everything I need to say is, and always will be posted at this web site, available to anyone interested. If some day it turns out that RSP in 'the arrow of spacetime' had hit the right track toward quantum gravity, I suppose people will be interested to explore it (see the download link above).

"When this lowly chap informed the Lucasian Professor of Mathematics that he had formulated the inverse square law of gravitation years before the publication of Principia, Newton is said to have flown into a rage. The two had already sparred over their optical theories, and when Newton took over as president of the Royal Society in 1703 (the year of Hooke’s death), he began erasing all traces of Hooke. Famously, he tossed the only contemporary portrait of Hooke onto a fire."

"An important scientific innovation rarely makes its way by gradually winning over and converting its opponents: it rarely happens that Saul becomes Paul. What does happen is that its opponents gradually die out and that the growing generation is familiarized with the idea from the beginning: another instance of the fact that the future lies with youth."

p. 1: "Equivalence principle implies no local definition possible in any situation: must attempt "regional" definition in regions at least as large as a wave-length.

p. 9:

p. 10: "A more elegant and potentially powerful scheme is to incorporate conformal techniques to bring null infinity to a finite point on the grid (emphasis added - D.C.), then can incorporate infinity into the computational domain (Friedrich, Husa, Lechner, Frauendiener all attending this meeting)."--------

You acknowledged in arXiv:quant-ph/0305088 that your "computer science students know very little physics". I am sure you know a bit more quantum physics than your students, and will be able to understand the argument at the link above, which answers the question in the subject line as well.

It seems to me that this UNdecidable state is protected from "decoherence", simply because it cannot reside in any Hilbert space. It isn't "observable", nor is computable in the sense of Turing (Topological Quantum Computation, 80.pdf). You can grasp it with your brain only.

May I take this opportunity to invite you and your colleagues to join our efforts toward a new relativity principle, outlined at

Note: Michael Freedman posted today (November 4, 2010) the 16th version of "Quantum Gravity via Manifold Positivity", arXiv:1008.1045v2 [quant-ph], in which he tried to obtain the dimensionality of space from the notion of 'empty set', and suggested a fleeting "pre-time" (ibid., p. 8). The latter emerges here from AOS.

Notice that the UNdecidable Kochen-Specker state at the first link above is a bona fide empty set from the viewpoint of its observed/actualized "projections" (if you like non-linear modifications of QM, try to attach this particular "empty set" to the manifold of states in geometrical formulation of QM).

To cut the long story short, "quantum computing" is impossible, even if it is "topological" and supported by Microsoft. Look again at the "general principle" in R.I.G. Hughes, The structure and interpretation of quantum mechanics, p. 77:

Firstly, this "general principle" does not apply to the case of UNdecidable, not-yet-physical, KS state, as explained by R.I.G. Hughes here. Secondly, because in QM we're dealing with operators, we cannot even imagine that some dynamical variable labeled with the anti-relativistic and "ideal Schrödinger time" (Jorge Pullin et al.) could possess some pre-existing values, like those we attach to each point of phase space in classical mechanics. Hence it is manifestly pointless to hope that we could control any of those quasi-local dynamical variables from the length scale of tables and chairs -- especially the Kochen-Specker state.

Now, if we look at GR, we have a similar puzzle with the equally incomprehensible "proper time τ " (Carlo Rovelli), in the sense that our wristwatch does read a continual line (1-D Euclidean space) of already linearized snapshots from the initial "proper time τ ". Looks like something is doing a linearized and flattening "collapse" of the "proper time τ " at each and every "sufficiently small" (Einstein) region of the points from our 'time read by a clock'. Hence the confusion in GR.

Not surprisingly, Mike Freedman didn't respond to my email (nor mentioned my efforts at the second link above). Due to the lack of interest, I will only reiterate the startling characteristic feature of 3-D space: finite things. If you wish to explain an elephant, all you need is to "obtain" its unique trunk; likewise finite things for 3-D space.

Details in my note on quantum gravity below. No other choice for self-wrapping of space and self-connecting of spacetime "points" seems possible. But first, one needs to identify an “instant”, such that it can be (i) “multiplied” and (ii) “ordered in time”, with “duration” equal to the infinitesimal separation of these instant(s), or rather one-multiplied-instant. More in Sec. Summary, pp. 35-36 from ExplanatoryNote.pdf.

All this is a tentative answer to Michael Teller's question (The Sunday Times, March 13, 2008): So long as the Universe had a beginning, we can suppose it had a creator. But if the Universe is really completely self-contained, having no boundary or edge, it would have neither beginning nor end: it would simply be. What place, then, for a creator?

Well, if The Universe does work like a brain, perhaps we may wish to consider an entity resembling our mind and consciousness. Say, [John 1:1].

Look at the following statement, from Stephen Hawking's "Grand Design" (to be published on September 9, 2010):

“Because there is a law such as gravity, the Universe can and will create itself from nothing. Spontaneous creation is the reason there is something rather than nothing, why the Universe exists, why we exist. (...) If there are trillions of universes as M-theory proposes, that luck and probability are enough to make our existence feasible, so no God was needed.”

Ignore the second and third sentences (spontaneous creation of an infinite multiverse doesn't make sense at all, or implies a very dumb and sloppy god).

How many factual and logical errors can you identify in the first sentence?

I think our understanding (S. Hawking, C. Isham, and myself included) of gravitation and the origin of inertia resembles my wife's knowledge in electricity, as she can comprehend Ohm's Law only by some analogy of water running in a flexible hose. As to Hawking's conjecture about "singularity", it was formulated as a theorem many years ago, and only after very specific and crucial presumptions, which do not hold in a world dominated by [we-do-not-know-it].

Yet Chris Isham will always praise his colleague, Stephen Hawking, and will never expose the factual and logical errors (non sequitur) in the first sentence above. Moreover, if the so-called Arrow of Space is correct, both his theory and Stephen Hawking's "Grand Design" will be wrong.But let's see the main ideas in the toposification of Quantum Theory, proposed by Chris Isham.

According to Gerard 't Hooft, “Isham believes another mathematical language may help, but I don’t think so. It sounds a bit as if describing the world in German is better than in Chinese.” Well, at least the "German" approach emphasizes on KS Theorem, which is usually obscured in the mainstream "Chinese" version of QM.

Chris Isham: "The interesting question really is, what do you mean by time?"

Isham believes that "every physical system, from atomic particles to the universe as a whole, can be viewed through different topoi" (source here), and suggests the notion of ‘pseudo-state’ (Würst); see Slide 28, from his January 2008 lecture "Topos theory in the formulation of theories of physics",

From Heidegger’s perspective, there is ‘no way things are’ in QM (Slide 13). My objection is not against Chris Isham's opinion but to its implementation: all topoi he can possibly design cannot exhaust/fully describe the "quantum trunk" rooted on 'the ideal monad without windows', because we are dealing with a new form of reality (dubbed 'potential reality'), which resembles Leibniz' monads and our cognitive structures, in line with 'the universe modeled as a brain'.

One important implication is that, on the one hand, the truth value associated with 'potential reality' is definitely YAIN -- not 'somewhere in between true or false' -- but on the other hand, Heidegger’s "things" are being explicated (as 'shadows on Plato's cave') in the quantum realm as reality 'out there' with unit probability, and their lifetime in the instant 'now' from the Arrow of Space is infinitesimal -- just a "point" from the continuum along the w axis (cf. Fig. 2).

Thus, we need two modes of spacetime, a global mode for 'potential reality' and a local mode for its fleeting explications along the Arrow of Space. Chris Isham is trying to place everything in one pot, while I separate them from the outset, and claim that their apparent fusion is due to the so-called speed of light, because the duration of their separation, in the local mode of spacetime, is zero.We have a perfect continuum of such explicated things -- one-at-a-time, along the Arrow of Space -- in the local mode of spacetime, at all length scales.

This is an absolute instant 'now' (cf. Isham's question above) from the Arrow of Space. It isn't physical, but looks like a transcendental tachyon, which is absolutely everywhere at 'no time' and at all length scales. It builds up the cosmological time and should not be 'GR observable', yet your wristwatch does indeed read it -- check out Luca Lusanna and the bewildered Tom Thiemann here.

The paradoxical situation is that I am strictly following Chris Isham's path to quantum gravity, announced in 1993. In my opinion, I am more 'Chris Isham' than he currently is. All differences boil down to the way he and I understand the continuum hypothesis and the quantum of action: dead matter makes quantum jumps; the living-and-quantum matter is smarter.

This is the motto of my web site, since July 1998. Let me try to explain it, by offering my version of 'the quantum principle'. Then I will try to answer the question about time posed above.

In simple words, the quantum principle is based on the Bootstrap Principle and the rule 'think globally, act locally'. It postulates the rules of the infinitesimal displacement of physical stuff in space and time, along the continuum of events produced by the Arrow of Space (bzw. arrow of spacetime): no "uncertainties" nor "quantum jumps" (verdammte Quantenspringerei, Erwin Schrödinger) exist in the quantum realm. We have continual trajectories of individual quantum particles, as well as emergent geodesics. The stochastic "quantum jumps" are artifacts from the measuring devices at the scale of tables and chairs.

If you examine the Gedankenexperiment with four dice, you will see that the transition from any n-state of the dice to the next n-state is perfectly continual in the local mode of spacetime: the duration of the EPR-like correlation "in the air" is zero in the local mode comprised from such n-states. The same holds for the emergent geodesics: the duration of the matter-geometry talk "takes place" in the global mode, at the interface of the non-Archimedean (geometrical) and Archimedean (physical) worlds, hence its physical duration in 3-D space (local mode) is zero. QM and GR are unified from the outset. Simple, no?

NB: If this quantum principle is correct, all other approaches to quantum gravity (Steven Carlip and Claus Kiefer included) must necessarily be wrong.

As to the question by Chris Isham above: 'time' is provided by the absolute clock of the Arrow of Space, which is external to all physical systems, yet is also "inside" each and every physical system, as it operates in the non-Archimedean realm of 'potential reality'. This absolute clock reads an infinite-dimensional time, which coincides with 'the time read by your wristwatch' (the dualist conception of time) only in the instant 'now' in which the global mode is being "flattened" to the local mode.

Of course, there is no way to find out if someone has found 'the right track', but at least I can think of QM & STR and the cosmological "constant", regardless of Chris Isham's opinion on my intellectual abilities and knowledge in theoretical physics.

As I'm still learning, since January 1972, perhaps some day I could say more on "the bridge"; check out the current version of my note on GWs at

Apart from that, I must acknowledge that Chris Isham is a very nice person. We met on November 13, 1998, and had many discussions in his Office. At our last meeting on March 9, 2006, he offered me a cup of tea, which was delicious.

I hope you can recall my email from Tue, 02 Sep 2003 15:51:53 +0300, regarding your arXiv:gr-qc/0308089v1 (cf. the subject line). Seven years later, I still don't know your viewpoint on the measurement (macro-objectification) problem and the clash of QM with STR (Abner has written a lot on this issue).

In your latest arXiv:1009.1220v1, you offered an alternative to the decoherence/einselection mess from Zurek, but the crux of the task has not been addressed: how do you reconcile QM with STR (Special Theory of Relativity)?

In order to reconcile QM with STR, we need 'reality out there', as suggested at the link above. Just drop "objective" and replace it with "potential".

As a bonus, I get your "translocal connections beneath the smooth surface of classical spacetime" (although not from your approach, of course), without any spukhafte Fernwirkungen or Gespensterfelder, plus much more,

Note: The so-called PR2 interpretationof QM stands for (potential reality) & (proper relativistic) interpretation1. It explains smooth bi-directional transitions between the classical and quantum realms, does not treat the former as some "limiting case", resolves the temporal solipsism of QM and recovers Bells' aether, and explains the actualization of potentialities as a relativistic process, in line with 'the quantum principle'. As a bonus, you get the physics of the human brain and all living creatures, without any "psi-fields" or other parapsychology: click on the smiling cat above.

Or don't. It's your free will choice, in line with the PR2 interpretationof QM.

I'm just a psychologist, don't need quantum gravity. Even if someone explains the origin of inertia and sorts out the quantum vacuum energy with exact equations, all this will be redundant information, just as I don't need to know the exact biochemistry of food processing in order to enjoy a beef steak, say.

Ch. 15, p. 136: "(W)e have no access to the past. We have only our 'memories' and 'records'. But these memories and records are in fact present phenomena. (...) The theory should account for the present correlations between these present phenomena....."The question of making a Lorentz invariant theory on these lines raises intriguing questions. For reality has been identified only at a single time."-------

1 The 'proper relativistic' interpretationof QM resolves the following problem, from Dah-Wei Chiou:

"The seemingly puzzle is analogous to the Einstein-Podolsky-Rosen (EPR) paradox, in which a pair of entangled particles are measured separately by Alice and Bob. In the context of special relativity, if the two measurements are conducted at two spacetime events which are spacelikely separated, the time-ordering of the two events can flip under a Lorentz boost and thus has no physical significance. Alice and Bob can both claim that the entangled state is collapsed by her/his measurement and thus have different knowledge about what the physical state should be(should have been - D.C.), yet the predictions by Alice and Bob are consistent to each other."

The UNdecidable state of the entangled/superposed "particle" (never in plural) exist as 'potential reality out there', and can always be traced back to the past light cone of both Alice and Bob -- retrospectively, after their "observations". Neither the "time-ordering" nor the "different knowledge" about what 'the physical state' should has been have any significance whatsoever to 'the base state', since the latter is omnipresent and can never be "collapsed". It is the 'back bone' along which Nature (not QM textbooks) has worked out smooth bi-directional transitions between the classical and quantum realms. Simple, no?

How wonderful that we have met with a paradox.Now we have some hope of making progress.Niels Bohr

Note: Please read Ernst Specker's 1960 article and notice [Ref. 1] above. To explain the seemingly mundane notion of 'counterfactual definiteness' from QM textbooks, and then compare it with the brand new situation introduced with KS Theorem (usually not covered in QM textbooks), consider an entangled quantum coin, which is being flipped "in the air". Upon landing on the floor (Hilbert space), there are "two" (in fact, one) observer(s), Alice and Bob, such that Alice can record the entangled quantum coin by viewing it 'from the bottom up', while Bob can see it only 'from above'. In such highly contrived Gedankenexperiment, Alice will know that if she observes 'heads', in the very same instant Bob should have seen 'tails' -- after all, it's one coin. Alice will also suppose that, by virtue of 'counterfactual definiteness', it is meaningful to ask: what would I had seen had Bob actually saw 'heads'?

I will spare the reader the usual excursion to Bell's theorem, EPR argument and its inevitable pitfalls (e.g., "quantum correlations happen without any time-ordering", and "nonlocal quantum correlations seem to emerge, somehow, from outside space-time", Nicolas Gisin), and will only stress that Alice&Bob are confined in the Hilbert space, hence can never "see" the quantum coin "in the air". The UNdecidable KS state shows up only in KS Theorem.

Look carefully at R.I.G. Hughes, p. 164, and notice the difference between the geometrical presentation -- in the 3-D space of the macro-world of tables and chairs -- of spin-1 vs. spin-1/2 systems. In the latter case, the component of the spin per se (a bare finger nail, cf. 'KS Theorem for teenage girls' above) can choose from two alternative observable values, either +1/2 or -1/2, both of which can be accommodated along one axis from 3-D space -- all they need is to choose a "diametrically opposed direction" (ibid.) along that same axis in 3-D space. Therefore, in the case of spin-1/2 systems, the UNdecidable KS state need not, and hence does not show up -- the 3-D space does not force it to reveal all of its contextualized classical-able states.

Not so in the case of spin-1 system: the square S2 of any component of spin can take three values -- 1, 0, -1 -- which in turn requires all three axes of 3-D space. In other words, even one square S2 will invade/require all available geometrical degrees of freedom provided by 3-D space -- the classical "filter" for contextualized classical-able states.

Now, Kochen and Specker have shown that, if you consider any triple of such squares S2 , each of which requiring the three mutually perpendicular directions in 3-D space (cf. Eq. 2.8 on p. 17 from C. Isham's textbook here), the statistical interpretation of QM would imply that you will always find out that "two receive value 1 and the third 0" (ibid.), similarly to the case of 'counterfactual definiteness' from QM textbooks. But the statistical interpretation of QM turn out to be wrong, as anticipated by Erwin Schrödinger in November 1950.

You can't have all the contextualized classical-able states in such {1,0,1} pattern, as proven by KS Theorem. Some of them will always and inevitably fail to comply with your (biased) expectation for "an unequivocal true-false value", as explained eloquently by Isham and Butterfield, p. 3 (see the excerpt above), and will have to be shifted into the UNcolored section from KS sphere (Helena Granström, p. 2). Hence my interpretation of the UNdecidable KS state above.

Notice also the logic of propositions regarding the "content" of potential reality, encapsulated with a single YAIN, and compare it to the topos approach by Chris Isham above.

I firmly disagree with the opinion in Wiki that KS Theorem (details in Mladen Pavicic et al., cf. p. 8 and p. 17) were some "complement to Bell's theorem". The latter theorem is based on counterfactual "reasoning", which might eventually be made clear and conclusive only in classical physics.

Regarding Bell's Theorem, Tim Palmer rightly noticed that (p. 7) "in order to establish Bell’s theorem, we need to consider correlations between pairs of measurements when the magnets have different orientations, let’s say n for the left-hand magnets and n' for the right-hand magnets. It is also necessary to assume that it is meaningful to ask: what would the spin of a left-hand particle have been had we actually measured it with magnets oriented in the n' direction (or, conversely, what would the spin of the right-hand particle have been had we actually measured it with magnets oriented in the n direction)?" Counterfactual "reasoning" is a recipe for parapsychology. More from Charles Tresser, Sec. 5.

There is no counterfactual "reasoning" in KS Theorem, ladies and gentlemen. Quite the opposite. Check out Ernst Specker above.

In summary, all quantum, as well as all gravitational "states" are contextualized quasi-local explications from their potential-reality state (never in plural): check out Fig. 2 above.

The Chinese Nebulae (located at the newly build National Supercomputing Centre in Shenzhen) achieved 1.271 PFlop/s running the Linpack benchmark, and can deliver a theoretical peak performance at 2.98 petaflops per second (FLOPS means floating point operations per second).

Suppose its Intel X5650 processors are "decoherent" quantum systems, such that all quantum fuzziness in the *timing* of their operations is "quite strongly peaked (notice the poetry - D.C.) about one path" (Jonathan Halliwell, arXiv:quant-ph/0501119v1).

Please calculate (roughly) how long your Chinese colleagues may run their Nebulae before it breaks down.

Please don't hesitate to publish your calculations.

I bet 100 EUR (sorry, no
US dollars) that *none* of the above will happen, for reasons explained at

The task is interesting to me because my brain has roughly 1014synapses, which do not make errors (I do; not my brain). Also, I cannot explain the generation of observable paths in Wilson cloud chambers, after Sir Nevill Mott (cf. Alessandro Teta, arXiv:0905.1467v1 [math-ph], pp. 9-10), with "decoherence". If people believe the latter is better than the Born Rule, they should explain (i) things we can observe, such as the generation of paths in Wilson cloud chambers, and (ii) things we cannot observe, such as 'global and absolute time' (A. Macias and H. Quevedo, gr-qc/0610057v1) along which space "expands". More on 25.11.2015.

I'm glad you offered a tentative verification of your general idea, that "even small violations at that scale can be magnified and lead to observable effects at large scales" [Ref. 2, p. 10] -- please see [Ref. 3].

Notice that any viable theory of quantum gravity must pass the reality check of 3-D space: "it is possible to look around, and see as far as we like" (Lee Smolin),

"According to general relativity, gravity is a characteristic of the structure of spacetime, so quantum gravity means quantizing spacetime itself. In a very basic sense, we have no idea what this means......"(For a nice review paper by Chris Isham on some of the conceptual issues in quantum gravity, go here.)"http://xxx.lanl.gov/abs/gr-qc/9310031

[Ref. 2] Idem, The Small Scale Structure of Spacetime; to appear in Foundations of Space and Time, edited by George Ellis, Jeff Murugan, Amanda Weltman (Cambridge University Press), arXiv:1009.1136v1 [gr-qc],http://arxiv.org/abs/1009.1136

p. 2: "Over the past several years, evidence for another basic feature of small-scale spacetime has been accumulating: it is becoming increasingly plausible that spacetime near the Planck scale is effectively two-dimensional. No single piece of evidence for this behavior is in itself very convincing, and most of the results are fairly new and tentative......p. 6: "For a small enough region of spacetime, one might guess that the causal structure is generic, coming from a random causal ordering......"We then face a rather bewildering question: which two dimension? How can a four-dimensional theory with no background structure or preferred direction pick out two “special” dimensions at short distances?.....p. 10: "There is a danger here, of course: the process I have described breaks Lorentz invariance at the Planck scale, and even small violations at that scale can be magnified and lead to observable effects at large scales [2]."

pp. 8-9: "The result indicates that there is no evidence so far of any quantum nature of spacetime above the Planck length. Spacetime there (distance of 7.3 billion light years from Earth - D.C.) is smooth and continuous."

To quote from the link above: “Because there is a law such as gravity, the Universe can and will create itself from nothing. Spontaneous creation is the reason there is something rather than nothing, why the Universe exists, why we exist. (...) If there are trillions of universes as M-theory proposes, that luck and probability are enough to make our existence feasible, so no God was needed.”

I have no questions to Professor Hawking regarding the second and third sentences, as spontaneous creation of an infinite multiverse doesn't make sense at all. It rather implies an utterly dumb and sloppy god (hence no capitals), which of course has nothing to do with [John 1:1].

The first sentence, however, clearly suggests that Professor Hawking holds strong opinions on the nature of gravity, quantum cosmology, and quantum gravity. Hence my question:

Q: With all due respect, do you realize that (i) you are a deeply religious person, obsessed by anti-theistic beliefs, and (ii) your book delivers many biased and unprofessional statements?

Should you disagree, please explain (i) the origin of inertia and (ii) your opinion on whether the quantum vacuum energy "gravitates".

> what is missing in my discussion of KS> (which is standard, see Bub's book)

I have quoted from it: see the letter by Schrödinger from November 1950 at the first link from my initial email.

> Please do not send me to any further texts by you, which are simlarly vague.

If you don't want to read and learn more, there is nothing I can do for you.

I wish you a pleasant and peaceful retirement.

Regards,

Dimi

> On Tue, Aug 17, 2010 at 9:13 PM, Dimi Chakalov <dchakalov@gmail.com>> wrote:>>>> Dear Petr,>>>> Apart from the "wellcome" typo in your abstract, the treatment of KS>> Theorem (p. 64) is incomplete, which drives your efforts to a blind>> alley: I cannot see how you could possibly reconcile your ideas with>> STR.>>>> Q: What is the "back bone" of your quantum state, such that you can>> make smooth, bi-directional transitions b/w the word of facts,>> governed by STR, and the quantum realm?>>>> My efforts, which you've been persistently ignoring, are at>>>> http://www.god-does-not-play-dice.net/#KS>>>> http://www.god-does-not-play-dice.net/#Levi_Civita>>>> If you cannot answer my question, check out the links above.>>>> I will appreciate the professional opinion of your colleagues as well.>>>> Regards,>>>> Dimi>>

You and your colleagues are hard-core Russians, and probably will not respond, as you never did so far. Even since August 1981, after my first effort to contact a Russian physicist (David Abramovich Kirzhnitz), I noticed this terribly rigid, Soviet-style thinking.

I also noticed that you are still unable to uncover the blueprints from relic GWs (p. 4 and ref. [4] in gr-qc/9907027). If you're interested in a fresh look at the task, feel free to reply to this email, after reading the text at the link above.

"This discovery of the last decade sharpens the classic horizon problem: why does radiation arriving from opposite ends of the Universe share the same temperature? The problem is now even more profound: how were the initial perturbations, with their puzzling synchronization, produced? What physical mechanism could have possibly planted these primordial seeds?"

If you use the 'spherical cow' (linearized) approximation of GR, the "puzzling synchronization" mediated by relic GWs will have to propagate in space, 'from one point in space to another', just like the z-direction toward the L-shaped tunnels of LIGO (e.g., arXiv:1007.3973v1, p. 11, Fig. 5). In the case of relic GWs, with "speed" faster than light.

If you drop the 'spherical cow' (linearized) approximation of GR, you will have to use pseudo-tensors to derive the magic L-shape of LIGO's arms. Try this:

Dimi Chakalov----Note: I read today the fifth version of a very intriguing paper by Den Yerokhin et al., Dynamics of Universe in Problems, arXiv:0904.0382v5 [astro-ph.CO]; 764 problems. My favorite one is about the "dark energy", p. 56, Task 9:

"Show that assigning energy to vacuum we do not revive the notion of ”ether”, i.e. we do not violate the relativity principle or in other words we do not introduce notion of absolute rest and motion relative to vacuum."

I emailed Dr. Yerokhin and said that the task seems impossible (I don't know how to define the stress-energy tensor of the vacuum in the first place, and then make sure that it is "proportional to the metric", J. Baez and E. Bunn), and added: "If you know how to solve it, please drop me a line with some hints and references."

The reply by Dr. Den Yerokhin (answer.pdf), along with his permission to post it on this web site, can be downloaded from Yerokhin.zip. You be the judge.

Regarding your latest essay: I know that you don't care about anything I suggest, as you never did in the past six years.

But since our brains are entangled, I do care about what I say or do *not* say to you. In the latter case, it would be bad if I don't tell you what I think about your latest arXiv:1009.1761v1 [gr-qc], although you will undoubtedly ignore it, since you're Russian.

There are two kinds of distances in the case of deons (Erast Gliner,arXiv:gr-qc/0006072v1): one is in the Riemannian spacetime of GR (examined in your arXiv:1009.1761v1 [gr-qc]), and the other is in the so-called global mode of spacetime,

The first distance is always finite, and the speed of gravity does not exceed the "speed" of light, while the latter distance is exactly zero, hence the speed of gravity there seems to be infinite, like a transcendental tachyon. With such dual speed of gravity, you get correlations *resembling* a school of fish:

You and your co-authors stressed that "the electromagnetic wave is unable to transport its own source, that is, electric charge" (arXiv:0809.2911v2 [gr-qc]; cf. p. 4 at the link below).

If I am on the right rack, the true gravitational waves (forget about quadrupole radiation) should be unable to transport their own source either, that is, the entity in the l.h.s. of Eq. 1, p. 35, at the link below.

Note: In addition to the comments on J. G. Pereira et al., arXiv:0809.2911v2, on p. 4 from ExplanatoryNote.pdf, see also Sec. Summary, pp. 35-37 therein, and another paper by J. G. Pereira et al., Does a tensorial energy-momentum density for gravitation exist? arXiv:0812.0034v1. Jose Pereira and his co-authors concluded that "at each point of the world-line, inertia compensates gravitation yielding a vanishing (bit not exactly zero - D.C.) spin connection" (p. 6), and elaborated as follows:

"This means that inertial and gravitational effects are both embodied in the spin connection [XXX] and cannot be separated because of the equivalence principle (notice that inertial and gravitational effects can indeed be separated inside each "point" of the emergent geodesic - D.C.)

"As a consequence of this inseparability, the energy-momentum current in general relativity will always include, in addition to the purely gravitational density, also the energy-momentum density of inertia. Since the latter is a pseudotensor, the whole current will also be a pseudotensor. In general relativity, therefore, it is not possible to define a tensorial expression for the gravitational energy-momentum density. This is in agreement with the strong equivalence principle which precludes the existence of such definition [3]."

If we define the energy-momentum current with the mass-energy conservation equation on p. 35 from ExplanatoryNote.pdf, we may recover the source of gravitational waves, placed in the l.h.s. of Eq. 1, p. 35 therein. It yields a time-conserved "charge" with only one "sign", and allows the cosmological "constant" to show up with different values along the non-unitary evolution of The Universe.

Let's start from scratch [Ref. 1]. I mentioned above that inertial and gravitational effects can be separated inside each "point" of the emergent geodesic, that is, in the postulated global mode of spacetime (not in present-day GR nor in the kind of teleparallel gravity studied by Jose Pereira -- read Janusz Garecki). The "point" in question is the one at which (or rather "inside" which) the two fluxes (Merced Montesinos) cancel each other completely, hence we may imagine some "locally inertial coordinate system in which matter satisfies the laws of special relativity" (S. Weinberg). To be precise, let me quote again Kevin Brown: "Einstein chose for his field equations a gravitational tensor whose covariant derivative vanishes identically, to ensure local conservation of energy-momentum, and this requirement is essentially equivalent to the geodesic hypothesis."

"However, it’s somewhat misleading to say that the equations of motion emerge from the field equations without having been imposed as a separate assumption. They follow as a direct consequence of the fact that particles follow “straight and uniform” inertial paths in each infinitesimal region of spacetime, and this in turn is a direct consequence of the
local conservation of energy-momentum. It’s true that the field equations of general relativity imply this conservation, as can be seen by
the vanishing of the covariant divergence of the Einstein tensor

"The field equations simply equate this to the energy-momentum tensor Tmn, so the covariant divergence of the latter must also vanish, hence energy-momentum is locally conserved,
hence particles follow geodesics.

"But the local conservation of mass-energy didn’t arise automatically, it was specifically designed into the field equations by the inclusion of the “trace” term (the term with coefficient 1/2) in the Einstein tensor. In fact, one of the early attempts of Einstein and Grossmann to formulate generally relativistic field equations led to the result Rmn = Tmn, but not surprisingly this is unsatisfactory, precisely because the covariant divergence does not vanish. After a great deal of searching (and with the crucial help from Levi-Civita - D.C.), Einstein finally realized that the natural conservation laws – and hence the law of inertia – is recovered if we include the trace term. David Hilbert arrived at this same conclusion almost simultaneously (in November 1915), although his route was much more direct, since he proceeded from a Lagrangian, which automatically leads to conservation laws.

"In view of this, it’s surely disingenuous to claim that the field equations of general relativity allow us to dispense with the independent assumption of equations of motion; the equations of motion essentially were imposed as a separate assumption, corresponding to the assumed conservation of energy-momentum that was intentionally built into the equations by the inclusion of the trace term.......

"It’s possible for the inertial mass of an electrically charged particle to be accelerated in a variety of ways while still conserving electric charge and current. In contrast, the conservation of energy-momentum fully constrains the motion of a massive particle (in the absence of non-gravitational forces), because the inertial mass is identical to the conserved gravitational “charge”. The principle of inertia is really just another name for the conservation of energy and momentum.......

"So, despite Einstein’s hopes, general relativity does not in any way explain or obviate the principle of inertia. Granted, if the field equations didn’t include the trace term (so that the covariant divergence didn’t vanish), the resulting theory would have many problems and be subject to many objections, but this goes without saying. No one disputes that the principle of inertia is extremely well-founded in observation. It is an extremely well-justified postulate – but it is still a postulate. General relativity does not explain inertia, nor does it dispense with the need to organize our spatio-temporal theories on the topology and morphology implicit in the principle of inertia and the associated distinguished coordinate systems."

[Ref. 2] A. Einstein, Autobiographical Notes: "In a gravitational field (of small spatial extension) things behave as they do in a space free of gravitation, if one introduces into it, in place of an “inertial system”, a reference system that is accelerated relative to an inertial system.".....

A. Einstein, 4 April 1955: "(T)he essential achievement of general relativity, namely to overcome ‘rigid’ space (ie the inertial frame), is only indirectly connected with the introduction of a Riemannian metric. The directly relevant conceptual element is the ‘displacement field’ (XXX), which expresses the infinitesimal displacement of vectors. It is this which replaces the parallelism of spatially arbitrarily separated vectors fixed by the inertial frame (ie the equality of corresponding components) by an infinitesimal operation. This makes it possible to construct tensors by differentiation and hence to dispense with the introduction of ‘rigid’ space (the inertial frame)."

"Postulate of locality: An accelerated observer (measuring device) along its worldline is at each instant physically equivalent to a hypothetical inertial observer (measuring device) that is otherwise identical and instantaneously comoving with the accelerated observer (measuring device)."

Bahram (arXiv:1006.4150v1 [gr-qc]) believes that "an accelerated observer may be replaced in effect (Sic! - D.) by an infinite sequence of hypothetical momentarily comoving inertial observers; mathematically, the world line of the accelerated observer is the envelope of the straight (presumably - D.) world lines of the corresponding hypothetical inertial observers."

I don't believe in teleparallel gravity, and would rather "insert" in that 'envelope of the straight world lines' a special mechanism, which could perhaps make GR *quasi-local*, as well as produce quasi-local positive mass in an asymptotically flat spacetime: please see my email to Dr. Mamdouh Wanas (printed below), and pp. 35-36 from Sec. Summary in

On Sun, Oct 10, 2010 at 11:14 PM, Friedrich_Wilhelm Hehl<hehl@thp.uni-koeln.de> wrote:>> Dear Dimi Chakalov,>> Thank you for your email. Of course, I am always open for discussions.> However, your ideas are so far from my actual field of interest that I> cannot see an overlap with your ideas.

----->> I wonder if you plan to elaborate on the postulate of locality, from>>>> Friedrich W. Hehl and Bahram Mashhoon, Nonlocal Gravity Simulates Dark>> Matter, Slide 3 at>> http://www.thp.uni-koeln.de/gravitation/mitarbeiter/nlGrav2010DPG1.pdf>>>> "Postulate of locality: An accelerated observer (measuring device)>> along its worldline is at each instant physically equivalent to a>> hypothetical inertial observer (measuring device) that is otherwise>> identical and instantaneously comoving with the accelerated observer>> (measuring device)."-----

There is too much poetry in this postulate. If we add gravity to matter, the latter will have to be *quasi-local*. Clarifying the exact meaning of *quasi-local* is the scope of my efforts. We just have to zoom on the infinitesimal, which is (poetically) considered to be "of small spatial extension":

A. Einstein: "In a gravitational field (of small spatial extension) things behave as they do in a space free of gravitation, if one introduces into it, in place of an “inertial system”, a reference system that is accelerated relative to an inertial system."

Stated differently, I think you've taken a wrong path marred with too much poetry.

I will appreciate the professional opinion of your colleagues as well.

F.W. Hehl, Spin and Torsion in General Relativity. II. Geometry and Field Equations, General Relativity and Gravitation, 5 (1974) 491-516

F.W. Hehl, On the kinematics of the torsion of spacetime. Found. Phys. 15 (1985) 451-471

As Asher Peres used to say, "these things were well known to those who know things well", and Friedrich-Wilhelm Hehl is definitely one of the people who 'knows things well'. Regarding his 1985 article cited above, there are plenty of physical reasons (e.g.,. Salvatore Capozziello et al., arXiv:gr-qc/0101038) to believe that the spacetime continuum, viewed microscopically, should carry a torsion, but the dynamics of torsion, which F.W. Hehl "left for a forthcoming article" (October 1984), is still missing. Meanwhile the so-called DDE showed up, and the dynamics of torsion became a highly non-trivial task: the spin density of matter is not the source of torsion.We should dig deeper than R. Penrose. Very briefly: the 720-degree rotational invariance of spinors may be interpreted as two "circles" resembling 8 , which pertain to the atemporal "handshaking" of two waves in the postulated global mode of spacetime; the end result "happens" on null-surfaces (Kevin Brown) "at p" [Ref. 1], and has zero duration ("small spatial extension"), as recorded with your wristwatch. The key ideas are that physical events (local mode) emerge on null-surface, and are inherently quasi-local, with vanishing (present continuous) torsion; details on November 25, 2015. Notice the linked text in [Ref. 1] and the notion of 'self force' in Machian gravity [Ref. 2], and check out Eq. 1 on p. 35 from ExplanatoryNote.pdf regarding the axiom of 'dominant energy condition'.

The null-energy conditions needed for singularity "theorems" and Tipler's theorem are wishful thinking in GR (references above). Because the "dark" energy from the quantum vacuum acts as an additional, all-permeating and perfectly smooth field, we encounter a blatant violation of Newton’s third law; check out an explanation with a car here: "The size of the force on the road equals the size of the force on the wheels (or car); the direction of the force on the road (backwards) is opposite the direction of the force on the wheels (forwards). For every action, there is an equal (in size) and opposite (in direction) reaction."

Compared it with the ultimate free lunch from DDE (August 2006): "Suppose you accelerate a car, but the gauge fuel shows that you're actually gaining more fuel by accelerating the car. That's the ultimate 'free lunch' provided by DDE, only physicists cannot explain it." It doesn't get diluted as space "expands". On the contrary, it's getting more and more. From the perspective of contemporary physics, it is far more shocking than perpetuum mobile, simply because it exists.

We cannot apply Newton’s third law (Hans Ohanian) to the source of DDE - some perfectly smooth stuff that "has zero inertial mass" and "can be accelerated with no cost, no effort" (B. Schutz). From the viewpoint of classical physics, this perfectly smooth [we-do-not-know-it] had its highest value at the instant of "inflation" (like the amount of fuel in a tank car), and then should have depleted rapidly, or in accelerated fashion during our current cosmological stage. Quite to the contrary: it springs out from "thin air" and does not, in any way, resemble the Casimir effect, say. It is a genuine non-unitary phenomenon, in the sense of John Wheeler's dictum 'Time is Nature's way to keep everything from happening all at once'. More in the papers above.

Perhaps the reader may wish to consider a fiber over a point p Є M as collection of spinors (cf. p. 4 from Geroch; general info here and here), and see if one can make the "fish" p more flexible: p should be endowed with Lorentzian metric only at some (still unclear) quasi-local positive-mass limit at p . The aim is to replace (i) the misfortunate splitting of spacetime in GR with an emergent spacetime along an arrow of space (compare it with R. M. Wald), and (ii) the misleading notion of "curvature" [Ref. 3], usually depicted with some "curved" 2-D surface immersed into 3-dimensional flat Euclidean space (John Baez), with atemporal correlations in the global mode of spacetime.Have you seen a school of fish? I will take off my Euclidean spectacles [Ref. 3] and try some reverse-engineering. An incomplete list of tasks is outlined below.

Look at the fiber bundle hairbrush at Wiki, and compare it with the "spinning" (along two "circles" resembling 8 ; see above) quantum hedgehog, which is supposed to show up upon zooming on the infinitesimal point p at Planck scale.

Q: Can you suggest a new nontrivial bundle (two Möbius strips maybe?) for the quantum hedgehog, which could facilitate the non-linear negotiation between an arbitrary quasi-local "fish", at some infinitesimal point p , and 'the whole school of fish', in Machian perspective? The symmetry operations related to 'dominant energy condition' (cf. above) should be defined over four sectors (notice the fav icon of this web site), in a very remote analogy with Kruskal-Szekeres diagram.

Clarifications: thequantum hedgehog has infinitely many "bristles", which should also facilitate the global view on 3-D space (Wiki): one would see "all points in 3-dimensional space simultaneously, including the inner structure of solid objects and things obscured from our three-dimensional viewpoint" (see below).

Also, the quantum hedgehog should somehow (sorry, I'm totally speechless here) resolve the main puzzles of (i) finite things in space and (ii) the "boundaries" of 3-D space with a new version of Finite Infinity. The assumptions here are that, depending on the "direction" we look at the global mode of spacetime, it will show up as either 'tending asymptotically toward zero' or 'tending asymptotically toward infinity', while a global (omnipresent) observer would "see" the whole 3-D space (local mode) simultaneously, en bloc. Namely, the duration of the glancing at 'the whole school of fish' (the whole 3-D space) will be zero, as recorded with our wristwatch, because the luxon-like dynamics of the two "waves" (cf. above) does not feel any physical time. It's atemporal. Locally, our global time is zero, and the re-created (AOS) local mode of spacetime is a perfect continuum. Hence all effects from the global mode are inevitably holistic ("dark", after M. Turner).Last but not least, the ultimate puzzle of the kinematics of space concerns the origin of the so-called "speed" of light: in the local mode of space, the "points" (quantum hedgehogs) of the underlying manifold are chained in such a way that (i) there is nothing between them, (ii) there is an upper limit on the speed of passing physical stuff 'from one point to the neighboring one', and (iii) there is a whole mirror world (Yakov Terletskii) on 'the other side' of this "speed" limit.

This "smuggling" can be explained by recalling that neither the coordinate time t nor the proper time τ (Wiki) along spacetime trajectories can be used as an independent variable, as τ is a "complicated non-local function of the gravitational field itself. Therefore, properly speaking, GR does not admit a description as a system evolving in terms of an observable time variable. (...) In the quantum context a single solution of the dynamical equation is like a single “trajectory” of a quantum particle" (C. Rovelli). Luckily, given 'the quantum principle' and the so-called PR2 interpretation of QM, one can design such 'single quasi-local trajectory of a single quasi-local quantum particle', in which the holistic input from 'the school of fish' is smuggled into the infinitesimal point p from the quasi-local trajectory of the quantum-gravitational "fish".

To be precise, at point pn the holistic input from 'the school of fish' is wiped out completely, by "cancellation of energy and energy flux of the real gravitational field with the energy and energy flux of the inertial forces field" (Janusz Garecki); hence the "ether" (global mode of spacetime) cannot show up (M. Montesinos). In the fleeting linearized "snapshot" at pn (local mode), the total energy of the gravitational field is always zero, the Einstein tensor and the energy-momentum tensor vanish identically, and all "dark" stuff has been linearized and physicalized. The same re-cancellation occurs at the nextpoint pn+1 along the Arrow of Space, but because these seemingly "neighboring" points belong to two different universes from the "flattened" local mode of spacetime (cf. Fig. 1 above), their fleeting physical content is different , while the holistic origin of this difference is "dark". Hence no "fish" can register any "deviation" (with respect to what?) during its re-created quasi-local geodesic: its "geodesic equation" has been updated dynamically, at each and every next point pn+1 -- a genuine Phoenix Universe (Georges Lemaītre, 1933). In the terminology of Karel Kuchar, this infinitesimal shift is mediated by the Perennial, which governs the dynamics "from outside as an unmoved mover". Notice that "all timeτis eternally present" (K. Kuchar), as it should be.

In general, if we agree that gravitational energy should be defined over finite volumes of space (e.g., B. Schutz), the first off mathematical task is to resolve the main puzzles of 'finite things' in space wrapped by its own "boundaries" at Finite Infinity (details above). This is what produces 3-D space per se, such that (operational definition) we can discriminate between 'inside vs. outside' and 'left vs. right'. These fundamental qualities of space are being wrongly treated as rigid background since 1915. If we introduce an Arrow of Space, perhaps we can recover the dynamics of GR: global time can only emerge from dynamical space.

The latter is defined with respect to the reference fluid in GR [Ref. 6, pp. 31-33]: it is the 'absolute structure' (James Anderson) of 'the universe as ONE' (global mode of spacetime), which bootstraps all quasi-local quantum-gravitational "fish" into a school of fish. Notice that the "chooser" (P. Pearle) of every next state of a given "fish", along its quasi-local geodesic, is the bootstrapping mechanism of 'the whole shoal'.

As I said above, I don't like miracles, like "decoherence" and "ideal Schrödinger time" [Ref. 7]. The calculations in QM and QFT explicitly presuppose a perfect clock endowingthe whole universe with some"global and absolute time" (A. Macias and H. Quevedo),but no physical stuff can reproduce it. It is an acute miracle, especially in GR. Physicists love to ponder on some Cauchy surface [Ref. 8], but it is a smooth differentiable miracle that cannot be assembled exclusively by physical stuff either (cf. Paul Tod's video lecture). Mathematicians used pure math and intuition to "assemble" these smooth differentiable miracles, then the founding fathers of GR (Levi-Civita, Einstein, and Hilbert) agreed upon introducing Lorentzian signature by hand, and banned these 'absolute structures' and Perennials.

That's what went wrong in 1915. We need new ideas about the origin of space. If you look carefully at the Finite Infinity and the pre-geometric plenum, you will realize that these are the only possible solutions for 3-D space to be wrapped by itself, and to produce a perfect continuum (see above): in the latter case, 'the universe as ONE' is being multiplied as infinitely many (uncountably infinite) infinitesimal "points"chained in the local (physical) mode of spacetime by 'the whole universe as ONE' ... which is in turn non-existent there!

This is the pre-geometric plenum made by the so-called Aristotelian Connection; all geometrical "points" from 'the grin of the cat without the cat' (Alice) are connected by themselves (local mode) -- one-connection-at-a-time along AOS. No other option for self-wrapping of space and self-connecting of points seems possible.

To explain the claim in the preceding sentence, let me elaborate a bit more on the self-wrapping of 3-D space with the drawing below, bearing in mind the equation LS = 1here. The meaning of finite things (elephant's trunk for 'space') is denoted with the unique number 1 , which can be seen above as well. Notice the interplay of Archimedean and non-Archimedean geometry, which is missing in diff geometry textbooks (e.g., Chris Isham's one here) and in present-day GR.

Once the size of the infinitesimal (S) reaches the non-Archimedean realm of geometry, it becomes a geometrical point p (from Planck scale), with "size" running asymptotically toward 0 , while the "size" of L is running asymptotically toward infinity, along the vertical axis. The global mode of spacetime refers to 'the whole universe as ONE' (Lucretius). The latter is totally missing in the local mode due to the "speed" of light, hence the physical space of 'finite things' becomes a perfect continuum: all points p are entangled and self-connected by their 'common cause' of ONE (global mode), and by virtue of LS = 1.No other option seems possible.

The drawing above can also explain the idea of dual cosmological age: finite in the global mode, and infinite/indecisive in the local mode, as the deflation time can never actually reach The Beginning at 0 ; details in pp. 35-36. As to the postulated Equation of Space, picture yourself riding the infinitesimal toward 0 : from the viewpoint of the global mode, one could imagine an "accelerated" shrinking stage by approaching "zero" size/The Beginning, but in the local mode such "accelerated" stage is an illusion. Likewise in the case of time-and-space-reversed "direction"; see the original full drawing here: people believe that space is "expanding", but it actually isn't, because it would take an infinite cosmological time (local mode) to actually reach 'the maximum large space', denoted with L .

There is no "accelerated" stage (Emil Mottola) in "shrinking" or "expanding" the volume of space by approaching asymptotically S or L . The confusion with the old (since 1930s) cosmological "constant" problem is due to the shape of spacenear the two "boundaries" defined with Finite Infinity, and on the unwarranted assumption that the "dark energy" is due to some [we-do-not-know-it] with positive mass density (an 'elephant in a china shop'), so it enters the current equations in GR: "a negative pressure can overcome a positive energy density" (ibid.). Alternatively, check out Eqs 1 and 2 in ExplanatoryNote.pdf, pp. 35-36.

(The very cracks, through which the physicalized (=converted into positive, cf. Eq. 1, p. 35) vacuum energy gets smuggled into the local mode of spacetime, vary in a wide interval, from producing "6 × 10-10 joules per cubic meter" (John Baez) to an equivalent in energy to 5 (five) solar masses emitted in under 60 seconds in the form of X-rays and gamma rays; all this "dark energy" comes from the "ether", ranging from an almost vanished flux to "1054 ergs/pulse" inGRBs.)

The 'physical size of lengths in 3-D space' (the scale factor) would have to actually expand if we were limited to Archimedean geometry only. Were that the case, one could eventually picture some conformal recipe for reaching infinity by "rescaling the metric", as envisaged by R. Penrose.

Regarding the "size" of 'the maximum large space', L , in the proposed version of Finite Infinity: look at the slope in the current "accelerated" stage from NASA, and map it to a reversed/inverted drawing (cf. the full original drawing here): can you extend the curve (not shown above) to reach an "accelerated" stage of approaching asymptotically infinite space? This will be the ultimate "cosmological horizon" for gravity. Because gravity cannot operate in infinite space, L should possess a numerically finite but physically unattainable value, perhaps in a manner resembling the "speed" of light for bradyons.

Finally, notice the Gedankenexperiment with an observer witnessing a "shrinking" bzw. "expanding" table with length two meters, starting from the macro-world. However, "It is very hard to imagine what new physics would introduce a cutoff on a scale of the order of 0.01cm" (L.H. Ford, gr-qc/0504096v2, p. 6). I am only suggesting that the dual notions of 'small running toward S' and 'large running toward L' are relative to the length scale of tables and chairs. Namely, a companion observer watching the "changing size" of physical objects will not notice any difference whatsoever: her table stretched to the size of a galaxy, and her table shrunk to the size of an atom, will always keep its "size" of 'two meters' (or "0.01cm", L.H. Ford). This is an effort to revive the old idea of 'mutual penetration of the Large and the Small', but it will take a lot of work to identify the new symmetry operations for 3-D space, starting from the macro-world, along some reversible time-and-space "direction" toward S and L .

I intend to elaborate on the time-and-space reversed "direction" (inverted space with its CPT symmetries, like inverting a rubber glove inside out) and the VGP formulation of GR on 25.11.2015. The full original drawing here offers some hints for interpreting the possible forms of 'mass' (Yakov Terletskii) and the adiabatic separation of positive and negative mass [Ref. 9], but much more work is needed to clarify the whole bundle of issues and the interpretation of "torsion effects".

"In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K(σp) depends on a two-dimensional plane σp in the tangent space at p ."----------

"These are the laws of an instant in canonical gravity. (...) In general relativity (notice Einstein's opinion here - D.C.), dynamics is entirely generated by constraints. The dynamical data do not explicitly include a time variable," says Karel Kuchar. Which is why the global time from the "expansion" of space is "dark" in current GR. In the forthcoming VGP formulation of GR, the "constraints" are produced by 'the shoal of fish' and are similar to the [10, 20] condition for emergent quasi-local geodesics (cf. below). Thus, the corrected QM and GR are united at the 'base manifold' of emergent continuum.

"In general relativity, the positive energy theorem (more commonly known as the positive mass theorem in differential geometry) states that, assuming the dominant energy condition, the mass of an asymptotically flat spacetime is non-negative; furthermore, the mass is zero only for Minkowski spacetime.'See also: Hans Ohanian, The Energy-Momentum Tensor in General Relativity and in Alternative Theories of Gravitation, and the Gravitational vs. Inertial Mass, arXiv:1010.5557v1 [gr-qc], http://arxiv.org/abs/1010.5557; pp. 2-4 and endnote 7, p. 32.

Footnote 5, pp. 5-6: "Locality, for which the metric tensor gmn reduces to the Minkowski metric and the first derivatives of the metric tensor are zero, is limited by the non-vanishing of the Riemann curvature tensor, as in general certain combinations of the second derivatives of gmn cannot be removed. Pragmatically, it may be concluded that violating effects on the EP may be negligible in a sufficiently small spacetime region, close to a given event......"the Apollo 15 display of the simultaneous fall of a feather and a hammer [4]......

Footnote 24, p. 21: "It is sometimes stated that the interaction of the particle with its own gravitational field gives rise to the self-force. It should be added, though, that such interaction is due to an external factor (...). In other words, a single and unique mass in an otherwise empty universe cannot experience any self-force. Conceptually, the self-force is thus a manifestation of non-locality in the sense of Mach’s inertia [135]."

p. 14: "In general relativity the problem of gravitational field energy is notoriously more subtle and complex. This is due to the nonlinearity of the field equations, which in turn is related to the fact that gravity carries energy and is thus a source of more gravity (emphasis mine - D.C.). In this sense gravity differs fundamentally from the electric field, which does not carry charge and thus is not the source of more electric field.....p. 17: "At present it is certainly not clear what might replace our present concept of spacetime at the Planck scale."

[Ref. 6] K. Kuchar, Time and interpretations of quantum gravity, in: Proceedings of the 4th Canadian conference on general relativity and relativistic astrophysics, ed. by G. Kunstatter, D. Vincent, and J. Williams, World Scientific, 1992.http://www.phys.lsu.edu/faculty/pullin/kvk.pdf

[Ref. 7] Jorge Pullin et al., A realist interpretation of quantum mechanics based on undecidability due to gravity, arXiv:1010.4188v1 [quant-ph]

p. 2: "Here we denote by t the ideal classical time that appears in the ordinary Schrödinger equation. (...) (I)f one make some judicious assumptions, namely, that the clock does not interact with the system, that the clock is in a highly classical state (a coherent state where the “hand” of the clock is sharply peaked in space and moves in a monotonous way), ... .....p. 3: "If one assumes one has a clock that follows the ideal Schrödinger time perfectly, ... .....p. 6: "The problem of macro-objectification of properties may be described according with Ghirardi [11] as follows: “how, when, and under what conditions do definite macroscopic properties emerge (in accordance with our daily experience) for systems that, when all is said and done, we have no good reasons for thinking they are fundamentally different from the micro-systems of which they are composed?”"

I will appreciate the professional opinion of your colleagues as well.

All the best,

Dimi------

Note: In order to pinpoint the "cracks" from 'the self-force' (A. Spallicci), through which the holistic ("dark") energy of 'the shoal of fish' gets smuggled into the quasi-local points of the emergent geodesic, check out S. Weinberg, p. 68 (links and emphasis added):

"Although inertial forces do not exactly cancel gravitational forces for freely falling systems in an inhomogeneous or time-dependent gravitational field, we can still expect an approximate cancellation if we restrict our attention to such a small region of space and time that the field changes very little over the region."

I will address (1) the localization of gravitational energy and (2) the notion of 'isolated system'.

2. See my comments to Bjoern Schmekel here. In order to fix energy densities at a "point", we need 'the whole spacetime' to be self-wrapped with "boundaries" along spacelike and null directions. It's a package. We also need some linearized and flattening "collapse", as I argued here; crucial details here, here, and here.

If you're interested, please read the text by following the links, and email me with your questions. Anything you weren't able to understand will be entirely my fault.

The most difficult puzzle to me is that our wristwatches are 'canonical clocks' that areat rest with respect to [we-do-not-know-it], and read our common 'cosmological time', so we have miracles in GR. Related to this puzzle is the long standing issue with the energy in the vacuum: it has an additional and unique freedom to stay in some "latent state", such that it may not contribute to "curvature", hence people can afford to look for 'energy differences' only (cf. John Baez' online paper here). If you measure the temperature of your bath tube, and find out that it as 'slightly above 37C', you can ignore such minuscule temperature increase (and publish your paper in Nature), but in our case we have two infinitely large and powerful sources of "cold and hot water", which cancel out their input almost exactly, to produce a minuscule observable increase 'slightly above 37C'. And this "slightly above" is manifestation of 'the ether'.

In other words, the "cracks" left for the holistic ("dark") energy of 'the shoal of fish' (see above) are incredibly flexible -- an upper limit, if any, on these "cracks" is totally unclear (cf. Eq. 1 on p. 35 from ExplanatoryNote.pdf).

All this unfolds from the textbook interpretation of those 'twice contracted Bianchi identities', as stressed by Hans Ohanian. Aren't you interested? Please feel free to disagree, and explain why.

Special Relativity must manifest in sufficiently small regions of his gravitational field and that these regions can be located anywhere in his gravitational field.

Ric = 0 describes a universe that contains no matter, by construction. But if that is so then there is no matter present to cause the gravitational field, bearing in mind that the field equations are also claimed to couple the gravitational field to its sources. One cannot remove matter by setting Ric = 0 (i.e. Tuv = 0) and then insert a mass, post hoc, by means of Newtonian two-body relations, into the resulting metric in order to introduce a cause of the alleged gravitational field “outside the body”.

General Relativity is a non-linear theory and so the Principle of Superposition does not apply. Consequently one cannot simply add masses to a given solution to the field equations. Every different configuration of matter requires a corresponding set of field equations to be solved. There are no known solutions to Einstein’s field equations for two or more masses and no existence theorem by which it can even be asserted that the field equations contain latent solutions for multiple masses.

Furthermore, point-mass singularities occur in Newtonian theory – they are called centres of mass. The centre of mass of a body is a mathematical artifice, not a physical object. Once can go to a shop and buy a bag of marbles but one cannot go and buy a bag of centres of mass of those very same marbles.

According to the astrophysical scientists it takes an infinite amount time for an observer to detect an event horizon. But nobody has been and nobody will ever be around for an infinite amount of time in order to confirm the presence of an event horizon. Consequently the concept has no validity in science. In addition, the aforementioned observer cannot be present in a spacetime that by construction contains no matter, assuming that observers are material. I do not see how an observer can be anything other than material.Concerning Einstein gravitational waves, none have been detected. This is also not surprising because the search for such waves is destined to detect none. Since Ric = 0 violates the physical principles of General Relativity Einstein’s field equations form an identity with zero so that the total energy of the gravitational field is always zero; so that the Einstein tensor and the energy-momentum tensor must vanish identically; so that the localisation of gravitational energy is impossible; and so that the field equations violate the usual
conservation of energy and momentum so well established by experiment. I wrote a paper on this which contains no mathematics:

That an erroneous theory can seemingly account for various observed phenomena is not new to science. The Ptolemaic system of epicycles accounted for various celestial phenomena but is nonetheless an erroneous theory.......

I now give you a simple recipe to prove me wrong. Prove that matter can be present in a spacetime that by construction contains no matter, prove that Einstein’s pseudo-tensor is not a meaningless concoction of mathematical symbols, and prove that r in Hilbert’s metric is not the inverse square root of the Gaussian curvature of the spherically symmetric geodesic surface in the spatial section of the Hilbert manifold.

"Q: How would you design a "geodesic" if the conditions for "conservation law" (non_conservation.jpg) are not fulfilled? I am referring here to the so-called "dark energy", which could spring from the l.h.s. of Einstein filed equation."

> Whether one considers Einsetin's covariant derivative on T_uv as a> conservation of energy law or as an energy transfer law seems to me> to be a matter of semantics and of no great importance. In any event> it is my view that Ric = 0 is inadmissible and so the Einstein field> equations violate the usual conservation of energy so well established> by experiment. The total energy of Einstien's gravitational field is always> zero and this is disastrous.

I am unable to relate your opinion to the question above. One obvious answer would be 'one cannot design such "geodesic" under these conditions', because, as M. Montesinos stressed in arXiv:gr-qc/0311001v1, pp. 4-5:

"More precisely, tµv = 0 tells us that the 'reaction' of the dynamical background metric is such that it just cancels the effect of 'flux' associated with the matter fields. It is impossible (and makes no sense) to have a locally non-vanishing 'flux' in this situation. If this were the case, there would be no explanation for the origin of that non-vanishing 'flux' (it will look "dark" - D.). Moreover, that hypothetic non-vanishing 'flux' would define privileged observers associated with it (the ether would come back!)."

I believe we all agree to keep the metric dynamical: it should keep playing its double role in the sense that it is both a field variable and defines the geometry *at the same time* (L. Szabados, private communication).

Then my approach to incorporating "dark" energy in GR is two-fold. On the one hand, keep the 'flux' vanishing up to 10^-122, that is, "the 'content' of energy and momentum densities and stress associated with the matter fields [psi] (which is characterized in Tµv) and the 'content' of energy and momentum densities and stress associated with the gravitational field [XXX]" (M. Montesinos) should cancel each other *almost* exactly.On the other hand, introduce a (brand new?) dynamics of this *almost exact* cancellation by two kinds of time (global mode and local mode) pertaining to *two standing gravitational waves*, such that we have *at the same time* (L. Szabados) an almost exact cancellation ą la John Cramer,

To be precise: the hypothetical 'two standing gravitational waves' pertain to the global mode of spacetime, and their perpetual cancellation re-creates the instants (plural) at which the negotiations b/w the two sides of Einstein equation ("space acts on matter, telling it how to move; in turn, matter reacts back on space, telling it how to curve", John Wheeler) are completed -- one-negotiation-at-a-time, along an Arrow of Space,

Please notice that the total energy of Einstien's gravitational field is tending asymptotically toward zero at each and every instant of *completed* negotiation, which is why people can introduce initially wrong assumptions (e.g., "vacuum equation" with Ric = 0 and linearized approximation of GR, to name but a few), although all these assumptions sound like the Born Rule in QM.

But of course you may have quite a different approach toward designing a "geodesic" under the conditions spelled out above. A penny for your thoughts!

I extend this request to your colleagues as well.

All the best,

Dimi-------Note: My efforts are too specific and may not attract the attention of experts in GR, so let me briefly explain why we need quantum gravity to understand GR. I will reproduce excerpts from the text offered by R. Aldrovandi and J. G. Pereira,

1. A real observer (§3.11) is not 'point-like', and therefore we need a quantum version of the strong equivalence principle (R. Aldrovandi, J. G. Pereira, K. H. Vu, Gravity and the Quantum: Are they Reconcilable? gr-qc/0509051 v1). This is a big can of worms -- check out the macro-objectification problem from GianCarlo Ghirardi, ref. [11] above. The solution proposed here is quasi-local emergent geodesic, resembling the trajectory of a fish -- it gets its nonlocal gravitational corrections and contributions from 'the whole school of fish', and follows locally a trajectory comprised of "points" -- one-correction-at-a-point along AOS. It makes no sense of talking about "curvature" (§3.81), because an ideal observer (§3.11) doesn't have Akasha-like memory to "recall" its experience without such corrections and contributions from gravity. The quantum-gravitational "fish" is a contextual and Machian-likerelational entity, hence it must never be "free from external forces" (§3.11).

2. You need (I don't) a constant gravitational field to make sense of "coordinate time" (cf. §3.36) and "finite distances in space" (§3.42), pertinent to "the scale factor R(t)", as defined above, with cosmological "constant" set to zero.

If, however, we have "time-dependent gravitational field, we can still expect an approximate cancellation if we restrict our attention to such a small region of space and time that the field changes very little over the region", says Steven Weinberg. There is too much poetry in this excerpt, as well as in other crucial ideas explained here.

The solution to the two tasks above is utterly needed. Do it, and if you come up with ideas different than those proposed here, I will consider my work redundant, hence wrong.

"This capacity of totalization, this gift of perceptive consciousness, is certainly one of its most important properties and unity may be the characteristic property of consciousness: consciousness unifies elements that otherwise are not related; from this comes meaning."

Can you read the text below?

Aoccdrnig to a rscheearch at Cmabrigde Uinervtisy, it deosn't mttaer in waht oredr the ltteers in a wrod are, the olny iprmoetnt tihng is taht the frist and lsat ltteer be at the rghit pclae. The rset can be a total mses and you can sitll raed it wouthit a porbelm. Tihs is bcuseae the huamn biran deos not raed ervey lteter by istlef, but the wrod as a wlohe. Pritie amzanig huh?

Voila !

I do hope some day we can get together. I deliberately missed some very important points at

May I share with you (and my son interested in KoWi) some thoughts prompted by your papers. I endorse the proposal for a third, Noetic world, different from res extensa and res cogitans [Ref. 1], which can be studied by empirical investigation, contrary to the claim by Edmund Husserl [Ref. 2].

Consider the Platonic ideas explicated with these sayings:

1. You can't hide a piece of broccoli in a glass of milk.2. Who has no horse may ride on a staff.3. Don't wear polka dot underwear under white shorts.4. Faute de mieux, on couche avec sa femme.

I'd say there are two such ideas. Here's a third Platonic idea, from Lewis Carroll:

'Twas brillig, and the slithy tovesDid gyre and gimble in the wabe:All mimsy were the borogoves,And the mome raths outgrabe.

p. 14: "... the horizons of meaning that we share (to different extents). This domain is not in the res extensa, but remains res cogitans. In other words, the meaning that we provide to the events does not “exist” physically, but incurs on us as one among a set of culturally possible meanings."

p. 15: "This communicative reality that the communicators shape over time and reflexively reconstruct cannot be considered as res extensa, but belongs to the res cogitans; it is not stable like matter, but remains in flux like language. It enables us to communicate in terms of uncertainties (e.g., possibly relevant questions) and expectations. Husserl (1929) recognized this realm as cogitatum, that is, the substance about which the Cogito remains uncertain. Our mentalpredicates provided to the world in intersubjective exchanges with intentional human beings, shape our culture and therewith ground what Husserl also called a “concrete ontology” or, in other words, “a universal philosophy of science” (1929, at p. 159)."

"According to Husserl, the study of this domain would provide us with “a concrete ontology and a theory of science” (ibid., at p. 159). However, the author conceded that he had no instruments beyond this transcendental apperception of the domain and therefore had to refrain from empirical investigation:

We must forgo a more precise investigation of the layer of meaning which provides the human world and culture, as such, with a specific meaning and therewith provides this world with specifically “mental” predicates. (Husserl, 1929, at p. 138; my translation).

>> I wonder if you agree with the core proposal for a third, Noetic world.>> I am not so sure that we need this proliferation. How many angels can> sit on a needle?

We're talking about an entity that exists as some form of 'reality', because the experiment I offered to your brain with the three Platonic ideas can be performed with all human brains, regardless of their structural differences, memory traces, life experience, age, etc.

The crux of the issue is that these *ideas* are (i) UNspeakable, (ii) robust and invariant to all differences in all human brains that have access to them, and (iii) open to newly created *ideas* that emerge in cosmological time. Regarding (iii), the idea of 'vehicle', for example, includes now a space shuttle, although 100 years ago it was still in the realm of 'the unknown unknown'.

> Perhaps, it suffices to distinguish between res extensa and res cogitans.> The Noetic world could then be considered as part of the cogitatum.

On the one hand, the Noetic world is sort of "part" of the cogitatum, to the extent to which it is revealed only by its association with our personal qualia. On the other hand, it is a form of reality (I call it 'potential reality'), because it exists 'out there' and evolves/unfolds from ... well, I'd guess from [John 1:1], or perhaps from 'the true monad without windows', or from the Noumenon, but thereis no way to verify such source.

There is no way to verify the ultimate source of res extensa either, yet we don't treat elementary particles as 'angels sitting on the pin of a needle'.

One final comment: if you consider the qualia of 'blue' (res cogitans), you can refer it to EM radiation with particular wavelength (res extensa), but if you consider any Platonic idea of the type from my initial email, the question is: what kind of 'potential reality' can facilitate its presence and faculties outlined in (i) - (iii) above?

Please bear in mind that my email correspondence and web site may be screened by those (almost) transparent guys from the Echelon, so we cannot enjoy privacy.

All the best,

Dimi-------Note: Some people can replace the Platonic idea of 'hot' with the one relevant to 'damp moss':

Is there a "bridge" connecting the quantum vacuum and the human brain? Perhaps all you need to extract ("dark") energy from 3-D space is right above your neck.

I've been arguing for many years that these Platonic ideas have dual ontological nature, because one can find their unique effects in both res extensa and res cogitans; and also that the need for quantum gravity stems from 'the universe modeled as a brain', as mentioned above.

NB: I have to clarify the precise meaning of 'dual ontological nature of potential reality', because we inevitably abuse this notion by using our language derived from the macroscopic world -- we tend to think about it as some sort of a "third" entity or "third world", which is "in-between" res extensa and res cogitans, and acts as a "bridge". But if you recall the notion of entanglement, it does not imply any "third" object or "bridge" connecting two parties. Instead, the "two" parties have become one inseparable object (ignore your classical thinking) due to their entanglement, such that we cannot factorize the "two" parts into some individual entities possessing well-defined properties apart from their entanglement (cf. quantum relational holism).

Likewise, the manifestation of 'potential reality' is two-fold, as we observe it in both res cogitans, in terms of Platonic ideas (cf. above), and res extensa in terms of pre-quantum KS state. The evolution of the latter is pre-correlated (Leibnizian pre-established harmony) with the former, hence there is no need for "anthropic principles" in explaining the "coincidence" or "Why Now? problem".

It is 'potential reality' what makes res extensa and res cogitansentangled by its dual ontological nature. Wolfgang Pauli explained his proposal in 1948; the initial idea belongs to Leibniz, of course. My suggestion from February 5, 1987 was to parameterize the phenomenon of entanglement, as manifestation of 'potential reality' in the quantum realm, with an open interval of real numbers (0, infinity), such that the latent observables (Henry Margenau) would acquire values larger than those pertaining to a macroscopic object (the latter has almost zero value of 'potential reality'). There should be a second layer of the quantum world, corresponding to greater values of entanglement, which will consume the possessed observables (mass and charge) as well, and will convert them into quarks; check out my LHC prediction from January 9, 2003. There should also be a very important, third section from this interval (mirroring the non-living macroscopic world), at which 'potential reality' approaches asymptotically infinity (cf. f(x) = oneOverX) -- the realm of Machian quantum gravity. At this "edge" of the physical world, the mutual penetration of the Large and the Small (global mode of spacetime) is almost entirely completed, and the physical world is approaching asymptotically 'the state of ONE'.

It is very difficult to thing about the human brain, with the human brain. If you're getting a headache, check out John Baez on duality; his interpretation is much easier to comprehend.

Their construction doesn't work, for a variety of reasons. Which is why I offer a radical approach toward 'the reference fluid' as "ether" -- an unobservable pre-quantum state of the whole universe as ONE -- and suggest a Machian quantum gravity.

> Personally, I think there are difficulties with either of these possibilities,> which I describe in my paper, but probably not everyone would agree with me.

I agree with you, and stress that there are even more arguments against Brown-Kuchar proposal.

> There would be a third possibility, which is to take a discrete set of> particles, so that only some points of spacetime can be labelled> explicitly.

But how would you make a 'discrete set of [whatever]' _and_ keep the spacetime continuum?

The task of such brand new quantization of spacetime is highly non-trivial, and goes back to the basic postulates of differential geometry. Which is why I asked you for advice, as I couldn't see even a hint for such task in your book.

My proposal is that "some points of spacetime can be labelled explicitly" thanks to a pre-quantum state derived from Quantum Theory,

Check out the classical limit of LQG -- you can do it now, no need to wait another 25 years -- by elaborating on the alleged continuum limit ("the refinement of the two-complex to take to infinity", Eq. 16), *in such way* that you could produce a classical 3-D space in which "it is possible to look around, and see as far as we like" (Lee Smolin, Three Roads to Quantum Gravity, p. 205).

There is no such possibility, Carlo. The requirements for observing 3-D space "as far as we like" will kill your LQG at its "continuum limit", Eq. 16.

On Thu, Dec 16, 2010 at 4:02 AM, Dimi Chakalov <dchakalov@gmail.com> wrote:>> Hi Carlo,>> I wonder what drives you to waste your time with "loop quantum gravity".>> I hear that many people at your university consider you an expert in> quantum gravity -- is this true?>> Can't you realize that "loop quantum gravity" is a pathetic joke? Check out>> http://www.god-does-not-play-dice.net/Margenau.html>> Please try to respond professionally, at least once. I know you can do it.>> Dimi>--------Note: Warning: If you do mathematics as a "bartender" and are not interested in the subtleties of 'finite space', don't waste your time by reading the text below.

Check out the continuum limit: this is the condition for 'finite space', which allows us to see through 3-D space "as far as we like" (Lee Smolin).

Now, try to implement this continuum limit condition in the alleged continuum limit of "loop quantum gravity" (LQG), or any other background-independent approach, causal dynamical triangulations (CDT) included. Carlo Rovelli won't even try, because he loves his LQG Barbie, but other people might be inclined to face the pitfalls of LQG.

There is a fundamental difference between all background-independent approaches to quantum gravity, published on paper, and the Machian quantum gravity (MQG), which is outlined at Henry Margenau's web page. It is about how we implement the continuum limit condition.

The perfect continuum of 'the grin of the cat without the cat', as observed exclusively by Alice, is interpreted in MQG as a special non-Archimedean state of the whole universe as ONE. This is the realm of 'geometry', presented with a hypothetical global mode of spacetime. It cannot be reached from the local (physical) mode of spacetime in the r.h.s. of Einstein equation, as demonstrated with the unresolved puzzle of 'continual finite space'.

With the exception of MQG, all approaches use some technique (more or less ingenious) to "zoom" onto the elementary building block of geometry, ensuing from a patch of finite space, after which they have to recover that samefinite space. In LQG, people arrive at the idea that the continuum of space might be comprised of some hypothetical stuff, say, some 1099 "atoms of volume" in every cubic centimeter of space, and try to explain how time "proceeds in discrete ticks of about a Planck time, or 10-43 second" (Lee Smolin, Atoms of Space and Time, Scientific American, January 2004, pp. 58-63).

Another idea is that there is nothing of that [whatever] stuff "between" the "atoms" of [whatever], much like "there is no water in between two adjacent molecules of water"(L. Smolin).

So far so good, but notice the crucial notion of 'nothing' in the concept of 'continuum': the consecutive points of a line mustsucceed each otherwithout any interval. That is, the "interval" is absolutely needed (otherwise all points from a line will fuse into one point), on the one hand, but on the other -- the "interval" must encapsulate the metaphysical notion of 'nothing' or "zero".

Now, how do you get this "zero thing", given its seemingly incompatible faculties? Can we 'have our cake and eat it'?

Except for MQG, people start with a patch of finite space, and then "instruct" something from that same patch to approach zero, e.g., some 'infinitesimal volume of space'. Look at Newton's recipe for obtaining 'the infinitesimal of time' (details from Robin Jordan).

Let's denote the infinitesimal of time with dt , along an axis z orthogonal to x/y, as in Flatland. dt is the infinitesimal time "interval" by which each and every finite (see above) timelike displacement is build along the Arrow of Space, as driven by the "dark" energy from 'the whole universe as ONE', exhibited with the alleged expansion of 3-D space, build by the infinitesimal space volume ds . Every physical system that can serve as 'clock' uses this elementary timelike displacement, dt . It is indeed a deep mystery, as acknowledged by Ted Jacobson (A Spacetime Primer, pp. 18-19), and it is hidden in the continuum.

To obtain the intrinsic space "interval", ds (the fluxions of x and y, after Newton, and the "atom of volume" in LQG), we again start from a finite (see above) displacement in space, and instruct it to approach zero. Notice that we inevitably use three points to obtain ds : the cutoff, s0 , always disappears from the local (physical) mode of spacetime, regardless of obtaining a finite interval (as in vacuum energy differences in QFT) or the infinitesimal ds .

Thus, we obtain ds by instructing s2 - s1 to approachzero. The same idea applies to dt . We believe that there should exist some numerically finite but physically unattainable "values" of ds and dt at Planck scale, but we know nothing about the ultimate cutoff s0 , because it is never present in the local (physical) mode of spacetime. We can only speculate about a special "boundary" by which 'space' is bounded at The Small, such that ds and dt match s0 .

This is the mystery of the continuum; sorry for repeating it all over again. Now, how do we implement the requirement that there should be "nothing" between the continual chain of consecutive points ds ? On the one hand, s0 must encapsulate the metaphysical notion of 'nothing' or "zero" -- the consecutive points ds of a line (1-D Euclidean space) mustsucceed each otherwithout any interval. On the other hand, we have to fulfill an equally important requirement that s0 must somehow exist in order to prevent all (uncountably infinite) points ds of a line to fuse into one ds . We need to 'have our cake and eat it', as sated above.

The metaphysical idea of 'continuum' is addressed in LQG with the mundane notion of 'zero something' (e.g., "there is no water in between two adjacent molecules of water",L. Smolin). For example, if you buy two bananas and eat them, you'll have 'zero bananas' in your hands. In LQG, the "bananas" are not converted into "zero" either, hence if you try to recover the continuum of finite space, you will ultimately fail.

Why? Because the remnants from the "bananas" will never, in no circumstances, allow you to recover the same patch of continual finite space. I will not delve into details about what constitute 'the remnants from the bananas' in LQG; think of these 'remnants' as the result from converting some characteristic of 'space' into something else, much like we explain (and convert) heat with kinetic energy.

Details available upon request.

Just a hint: because we don't live in some privileged or unique place in the universe, consider a finite volume of space with radius 7.3 billion light years (cf. Yuan K. Ha), which amounts to radius of 690.1025 cm and volume of 1,376,055,281.1025 cm3, each of which contains 1099 "atoms of volume". We have 1,376,055,281.10124, or roughly 10133 "atoms of volume" in which we can (i) picture 3 orthogonal dimensions of space, and (ii) see "as far as we like" (Lee Smolin). But because LQG is inherently relational, background-free theory, there is nothing at our disposal to 'hold onto', to avoid all errors in the binding of these10133 "atoms of volume".

In MQG, the notion of "zero thing" is not 'zero something' -- the hypothetical 'global mode of spacetime' at s0 is not present in the patch of continual finite spaceeven at the mathematical level of 'manifold'. From the viewpoint of the local (physical) mode of spacetime, it is 'zero nothing':

It is General Relativity itself which requires the global time-as-change from the Arrow of Space to be non-existent in the local (physical) mode of spacetime, or else the ether, as 'the reference fluid of GR', will show up in GR. The global mode of spacetime does exist, because otherwise you cannot "connect the dots" as consecutive points of our 3-D space, in such a way that we can see "as far as we like" (Lee Smolin). You can't bind the consecutive points of 3-D space with some "thermal time" (C. Rovelli, arXiv:0903.3832v3 [gr-qc], p. 8). Which is why the energy of the Arrow of Space is inevitably "dark" -- there is no way you can trace back its source from/within the local (physical) mode of spacetime. All you can observe in the local mode is spatiotemporal coincidences, but you cannot recover the phenomenon which literally builds up the 3-D space of GR, from GR. Which is why I tried to contact Professor Rovelli in November 1999. But of course he didn't reply. Physicists don't like to be reminded of Aristotelian First Cause -- it will also look "dark" in present-day GR. Which is why we need quantum gravity.

In another email, Carlo Rovelli stated the following: "All of us keep looking around, reading, checking out the papers in the archives, and when we find ideas that seem interesting to us, or potentially good, we react. There is no shortcut to that."

It is like refusing to read the news on your mobile phone, because news must be either displayed on TV or printed on paper. That's how Carlo Rovelli implements his rule of 'scrupulous intellectual honesty'. He will keep preaching to the choir and publishing papers on LQG in the next twenty-five years, and will probably end up like Alain Connes.

This whole story is anything but 'news', as it can be traced back to Lucretius, some 2060 years ago. However, as John Coleman rightly noticed, "it is extremely difficult to induce penguins to drink warm water."

The results are indeed laughable, like the chewing over how time "proceeds in discrete ticks of about a Planck time, or 10-43 second", and what might have happened "shortly after the Big Bang, when the universe expanded by 1028 within just 10–36 of a second."

Dear Dimi,thanks for your last email. I am just writing to let you know that after it I have added your name and address to my SPAM filter list. Therefore I am not going to see your mails anymore.[snip]

=================

Addendum

Pity Carlo Rovelli chose to run away.

For those interested in his viewpoint on LQG in arXiv:1012.4707v2 [gr-qc], let me offer just two excerpts (emphasis and links added) and my comments:

p. 7: "Meaning of quantum mechanics.

"Hopes have been voiced that a quantum theory of gravity will clarify the mysteries of quantum theory. This is not the case of loop gravity, which uses standard quantum theory (in whatever interpretation is your favorite one), only slightly generalized to make room for the peculiar way temporal evolution is described in general relativity."

The same C. Rovelli stressed in his gr-qc/0604045, p. 4: "The proper time [tau] along spacetime trajectories cannot be used as an independent variable either, as [tau] is a complicated non-local function of the gravitational field itself. Therefore, properly speaking, GR does not admit a description as a system evolving in terms of an observable time variable."

I do not know how "standard quantum theory" and LQG have been "slightly generalized" to "make room" for the absence in GR of "an observable time variable." Perhaps C. Rovelli is "slightly" confused with some slightly "partial observables". Well, it happens.

To understand the difference in the interpretations of "standard quantum theory", and their implications for LQG, see

"The “problem of time” is not anymore a conceptual problem in quantum gravity since the conceptual issues have been clarified, but remains a source of technical difficulties. The problems can in principle be solved using the relational formalism. That is, defining observables not with respect to unphysical space time points but in terms of relations between dynamical fields. (...) One possibility of constructing relational observables is to couple the theory to effective matter fields and use these as reference systems, in order to formally circumvent the difficulties deriving from general covariance [80]."---[80] C. Rovelli, “Quantum Reference Systems,” Class. Quant. Grav. 8 (1991) 317–332.

Rovelli's claim that "the problems can in principle be solved using the relational formalism" is nothing but wishful thinking -- check out Karel Kuchar's research (references at this web site). Of course, he does not offer in arXiv:1012.4707v2 [gr-qc] reference to any research paper published by Karel Kuchar.

"The exact treatment of the constraint is the central (as yet open) problem in loop quantum gravity."

Why is that? Because LQG is stuck at its kinematical stage. To obtain the dynamics, the first off task is to make sure that, in the classical limit of LQG, we will be able to see "as far as we like" (Lee Smolin).

> If you call my (and other's) work a "pathetic joke", I just turn the> page and read something else: I am not interested in a dialogue> based on insults.

Okay, I admit it was too harsh. Sorry. Let me please correct my statement: I think loop quantum gravity (LQG) is indeed a joke, but a smart one. It isn't 'not even wrong', because one can learn a lot about quantum gravity by refuting LQG.

I also wish to say that, from all pupils of Chris Isham, I consider you the smartest. But you shouldn't scoffed at Karel Kuchar's Perennials with those "evolving constants" and "partial observables", because nobody can beat Karel on GR. He showed you where you went wrong, with very simple math (you know the reference, right?), but you didn't take notice.

You did not reply professionally to any of my email messages sent since Fri, 26 Nov 1999 (printed below), and I'm afraid you will never reply to my criticism of LQG posted at

I am trying to comprehend how you [Ref. 1] and Prof. A. Ashtekar are trying to 'isolate time' or 'bring time' into quantum gravity. May I ask a question?

Since 1986, the main hope of the Ashtekar approach is that it may yield solutions to its own analogue of the Wheeler-DeWitt equation. This analogue involve functions of spin-connections, or loops in 3-space, and hence the picture is totally 'self-acting'. It is like Barron von Muenhausen who tried to lift himself and his horse by pulling himself up by his hair. In other words, you can't 'hold' on anything, for there is no background there.

If so, what could possibly 'bring the time' that we can measure with clocks from "spatiotemporal coincidences only" [Ref. 1]?

With kind regards,

Dimiter G. Chakalov[snip]

[Ref. 1] Carlo Rovelli. The century of the incomplete revolution: searching for general relativistic quantum field theory. Sun, 17 Oct 1999 19:43:18 GMT,http://xxx.lanl.gov/abs/hep-th/9910131

"Nature, apparently, always “turns on interactions” in a very special way."....Footnote 6: "Here, again, we are ignoring the diffeomorphism freedom, which, again, does not materially impact the present considerations."

[Ref. 2] R. Y. Chiao, How the conservation of charge can lead to a faster-than-c effect: A simple example, arXiv:1012.0797v1 [gr-qc]

"This implies that at the quantum, microscopic level of description, the disappearance of an individual electron, such as at point A, must always be accompanied by its simultaneous reappearance at an arbitrarily far-away point on the surface, such as at point B, at exactly the same instant of time [6]. Otherwise, the principle of charge conservation would be violated at the quantum level of description of individual events.

"We shall call this counter-intuitive effect "instantaneous superluminality due to charge conservation." Note that this superluminal effect does not violate relativistic causality because ...... ....."To sum up, charge conservation leads to situations in which causal, faster-than-c effects can occur. At the quantum level of description, such effects can lead to causally superluminal charge and mass currents inside matter."

> The crucial point is that analysed in a static laboratory frame the galaxies> do not appear to be accelerating or decelerating.> Therefore nothing is pushing them or pulling them.> In this sense there is no dark energy nor gravity out there although gravity> acts locally because of inhomogeneties.

Please correct me if I'm wrong.

You are implying a preferred reference frame of 'the whole universe', in which one can imagine non-intersecting geodesics converging towards the past, after Weyl's Principle,

In my opinion, such approach will produce mutually exclusive hypotheses, because this unique reference frame is both real and unreal. There are plenty of evidence in support of it (my favorite case is the cosmic equator, e.g., Craig J. Copi et al., arXiv:astro-ph/0605135v2), yet we just can't stick to it, because itrefers to something that is 'absolutely everywhere in no time'. We just call this entity "space".

If we take the Hubble Law literally, one option is to picture this 3-D space as some infinitely large warehouse, in which some "ambulances" are running away from any arbitrary "point" in the warehouse. Some astronomers really believe this may be the case. I personally consider it too primitive and brutal for our elegant and smart Mother Nature. I think She wouldn't have chosen the other option of "expanding" metric either, because it is also brutal.

So, I need to understand these two alternatives, bearing in mind that I may encounter mutually exclusive hypotheses, in which case I have to make a new one (which I did).

> If you think that dark energy acts upon the metric, not upon the matter, that> is another story.

I think the so-called dark energy is an artifact from our incomplete knowledge of gravity; please check out Addendum at

> Clearly in a normal static laboratory metric without expansion there is no> such thing.

It shouldn't, and can't show up there.

> So my analysis does not reveal anything about your metric.>> As I understand Einstein's equivalence principle, gravity can be replaced> by an accelerating frame of reference.

I haven't been able to understand Einstein's equivalence principle in the past 38 years.

> So for calculations on earth you can use a frame of reference that is> accelerating radially outwards.> This leads to the gravitational redshift etc.> Einstein did not ask what was pushing such a frame of reference outwards.

Angels, obviously :-)

Richard Feynman, Character Of Physical Law, p. 8: "The next question was - what makes planets go around the sun? At the time of Kepler some people answered this problem by saying that there were angels behind them beating their wings and pushing the planets around an orbit. As you will see, the answer is not very far from the truth. The only difference is that the angels sit in a different direction and their wings push inward."

> Surely it is just a mathematical trick: not supposed to represent reality.

> One of Einstein's many fruitful thought experiments.>> NOW this frame of reference has morphed into "space". Space they imply> is accelerating outwards from the earth and has been doing so ever since the> earth was born. By now space must be going very fast indeed. The earth is> no longer a source of gravity, but a source of "space".

> In my paper I try to understand the kinematics of the SN1A redshifts without> any preconceptions about gravity or space, just extending laboratory physics> to these distances.

I'm very glad that your paper was published.

> To my surprise the data fits very well.

Yes, sure, but the whole issue is *very* tricky, after the freedom in GR to manipulate gravity 'by hand' and with "a simple static cartesian frame of reference with no curvature and use special relativity," as you put it.

> What should we conclude ???? At least that dark energy and expanding space> is not proven or not the only game in town.

Not proven and not the only game in town.

> For me, "space" has replaced the 19th century concept of the luminiferous> ether.> Nice idea, but you cannot see it, detect it or measure it.> It is better to work with what we can observe.

And keep an eye on what we cannot in principle observe: an omnipresent, hence "dark", entity. Details in the web page of Henry Margenau above.Pity Martin cannot take part in our discussion. He said once that he has enough confidence in the multiverse to bet his dog's life on it,

> ________________________________> From: Dimi Chakalov <dchakalov@gmail.com>> Date: Tue, 4 Jan 2011 03:13:07 -0000> To: "Farley F." <F.Farley@soton.ac.uk>> Cc: Baron Rees of Ludlow <mjr@ast.cam.ac.uk>> Subject: arXiv:1005.5052v2 [astro-ph.CO], "There is no gravity> out there and no dark energy.">> Dear Francis,>> I couldn't find the sentence quoted in the subject line in your> abstract in Proc. R. Soc.>> If you adopt "a simple static cartesian frame of reference with no> curvature and use special relativity", under the assumption that "the> galaxies are receding from Earth at unchanging velocities in a fixed> metric with no curvature", how can you possibly find out that (i)> there is gravity out there and (ii) dark energy as "expansion" of the> metric? The latter will be totally invisible to you, hence your claim> in the subject line is not justified.>> Please correct me if I got your idea wrong.>> Regards,>> Dimi>----------

Note: Regarding the kinematics of the SN1A redshifts (the "ambulances") above, notice a tacit, and erroneous, presumption that one can expand a finite volume of space to 'the whole universe', and apply the interpretation of redshift effects from the former to the latter, assuming that some unique reference frame may offer a bird's eye view on 'the whole universe'. The same anti-relativistic error is made in NASA's drawing of 'the cosmological time' en bloc, after assuming some unique reference frame for the Weyl Principle. The simples explanation of this error is to examine carefully the misleadingballoon metaphor, bearing in mind that we deny the physical existence (local mode of spacetime) of both "the center" of the balloon and some 4-D spatial "direction" pointing to some not-yet-occupied space, waiting patiently for the universe to expand into. Instead, we keep them in the global mode of spacetime, which is assumed to be a pre-geometric plenum located "between" the points of the spacetime manifold, in which a torsion-like "loop" or "handshaking" is performed -- totally hidden by the "speed" of light.

But if you subscribe to the "block universe", nothing could help you identify the error of assuming such unique reference frame. Unless you try to define an 'isolated system in GR', to address the energy balance of 'the whole universe' and speculate about its "dark energy", you may never notice the anti-relativistic error. So, try to define 'isolated system in GR', with "dark energy". Try any conformal or you-name-it recipe, your choice. Can't make it.

Why? Because what makes the energy from empty space "dark" is the faculty of the whole universe to act on itself. It's self-acting, after Aristotle. Physically, only Baron Munchausen has managed to do it.

Alternatively, you may wish to start with a modification of G F R Ellis' FI. Or keep quiet, like Sir Martin Rees and his dog. It's your free will choice.

"The cosmologist Martin Rees, former president of the Royal Society, has won this year's £1m Templeton Prize – the world's largest annual award given to an individual. He was awarded the prize for his "profound insights" into the nature of the cosmos that have "provoked vital questions that address mankind's deepest hopes and fears"......."As the author of more than 500 research papers, ... (Comment: If Martin Rees has started publishing research papers at age 18, for 50 years he should have produced on average over 10 research papers per year -- D.C.). ......

"He adds he had no qualms about accepting the award and that has not yet decided how he will spend the £1m prize money."

Probably by offering even deeper insights into the nature of the cosmos and provoking the ultimate vital questions that address mankind's deepest hopes and fears from ... the multiverse ? Just guessing.

You have assumed that "the dark energy may be parameterized using an equation of state w(z) = P/p, where P and p are the pressure and dark energy density, respectively, and w(z) is an unknown function of red shift."

Are you assuming some absolute space in which objects recede with respect to any (=not privileged) location with *actual* velocities, in line with the Hubble Law (e.g., Michal Chodorowski, A direct consequence of the expansion of space? arXiv:astro-ph/0610590v3) ?

Or are you assuming that the metric, or perhaps the scale factor, may vary in such way that we perceive redshifted light from these objects, depending on the distance from us, but without *actual* velocities associated with the redshift effect?

I endorse this latter case, because the former doesn't make sense to me, as it implies some absolute space in which "more and more space appears" due to DDE.

Please advise.

Kindest regards,

Dimi Chakalov------

Note: Suppose the scale factor changes as function of the distance, like this:

Observer A will see redshifted light from some objects receding from her with increasing "speed" proportional to the "increasing" volume of space, as she perceives a galaxy as many times "larger" than her 1 "meter" table. Observer B won't notice any difference to the "size" of her table, because a galaxy will be 1 "meter" to her. In other words, Observer B will be seen by Observer A as "very tall and fat", i.e., stretched to the size of a galaxy, while Observer B will only notice see that Observer A has shrunk to a tiny little table of 1 "meter".

Who has "the right meter" ? Nobody. This is the essence of
Relative Scale Principle. It is applicable only to 'finite things'. There are two images from 'the universe as ONE' which is indeed absolute. It is is placed at the two opposite ends of 'space', which is why we see "them" as zero and infinity.

Notice that the "number" of points in the two cases above is the same: uncountably infinite (non-denumerable). This is the non-Archimedean world of pure geometry -- the grin of the cat without the cat, as only observed by Alice.

It goes without saying that nobody is interested. People keep arguing about the "expansion" of space "during" some "inflationary stage", as well as actual velocities of objects, as inferred from their redshifted light + Hubble Law, as if they have been sitting in some ideal/meta-observer reference frame at which they can take a bird's eye view on the whole universe en bloc, and measure some absolute dimensions of the universe. You need the reference fluid of GR.NB:People, let's make it clear, once and for all. The error you've been making is very simple: if you take the stand of an observer placed in 3-D space, such that there are finite volumes of space that you can identify -- inside vs outside or Small vs Large -- you cannot, not even in principle, extend this viewpoint to some absolute "bird's eye view" to include 'the whole universe', as with the NASA drawing, nor can you "see" the alleged "curvature of space".

If you could somehow reach the non-Archimedean realm of 'geometry', you may be able to see the whole universe en bloc, as depicted in the drawings below.

It doesn't matter if you step back 1 m, or move 1 m closer to the universe -- you will see the same universe en bloc, because you're in the non-Archimedean realm of "infinity" (recall the 'Aleph-null bottles of beer on the wall'here). Only Chuck Norris has been there (twice).You can't.

If you insist on talking about some redshifted "ambulances" receding from you with speed proportional to the distance, after the Hubble Law, keep in mind that you are again confined into the finite 3-D space -- left vs right and Small vs Large -- hence you can say NOTHING about the phenomenon which createsthisfinite 3-D space. You can only see its END result -- a finite 3-D space. You cannot see or "measure intrinsically" any "curvature" of spacetime nor "DDE of [whatever]". You cannot see or "measure intrinsically" any GLOBAL parameter of spacetime.

In the framework of present-day GR, you cannot have any privileged class of "fundamental observers" (Jean-Philippe Uzan) nor absolute coordinates of Earth in the absolute reference frame of the cosmic equator (Craig J. Copi et al.). To be precise, "in Friedmann-Robertson-Walker solutions one has canonical clocks (e.g. the temperature of the cosmic background radiation) that not only break Lorentz invariance defining a cosmic (global) time but break the Galilei invariance defining observers which are at rest with respect to the cosmic background radiation" (Luca Lusanna et al.). The current GR can't explain the bold fact that your wristwatch does indeed read this "unphysical" (global) time (Thomas Thiemann).

The solution is simple and non-trivial: dual age cosmology. That is, the age of the universe is finite in the global mode (currently some 13.7 billion years "after" The Beginning), and infinite/indecisive in the local mode of time, as read by your wristwatch. Once created by [John 1:1], the universe has "already" become eternal/infinite along both directions of the local mode of time, toward the past and the future.

Again, in present-day GR, you can't define any external and absolute parameter to map the END result from DDE to some cosmological "timeline", so that you can propose some "equation of state" (EOS) of DDE. It is just as wrong as are the following statements ("Big Bang 'soup recipe' confirmed," by Rolf H. Nielsen, New Scientist, 11 June 2003):

"A microsecond after (absolute time - D.C.) the Big Bang, when the exploding fireball of the newborn Universe was only a few kilometres across (absolute space - D.C.), all matter existed in a special state."

Such statements about the age of the universe are not better than the "discovery" of the Archbishop of Armagh James Ussher that the Earth was formed at 6 p.m. on 22 October 4004 BC. Only Chuck Norris can, at least in principle, measure the absolute age of the universe, but hasn't yet publish his calculation.

Wilma, did you notice how fast was the latest inflation? And look at the scale factor: it changed that much by the time I finished my coffee!

Dimi Chakalov----Examine the objects as they are and you will see their true nature;look at them from your own ego and you will see only your feelings;because nature is neutral, while your feelings are only prejudice andobscurity.

Shao Yong, 1011-1077------------------

Note: The manuscript by Miao Li et al. [Ref. 1] is the best review of the "dark" puzzle I've ever read. It is exceptionally clear and well organized. It is a joy to read and study.

My email above was prompted by the statement on p. 5: "The null energy condition is marginally satisfied." It will be nice if they elaborate on the adverb 'marginally'.

Basically, the energy conditions express the idea that the locally-measured energy density must be strictly positive everywhere and for all observers. Sounds trivial, like saying that the mass is always positive, right? Well, try to prove it.

The catch is in the presumption of locally-measured energy density. The "measurements" are actually quasi-local due to gravity. "One cannot just integrate Tuv(Matter) over a 3-space to obtain a conserved total energy-momentum" (R. Penrose, 1966). Moreover, this whole 3-space has its own dynamics, but one cannot use any external absolute background time to write down EOS of the "dark" puzzle.

As Lau Loi So et al. acknowledged in their study with classical pseudotensors, "one can regard positivity as an important test for quasi-local energy expressions" (arXiv:0901.3884v1).

And the other way around. It's a bundle. But we cannot use pseudotensors. It is a bit as if you're doing an exercise in analytical chemistry, and are trying to prove that you have NaCl in your sample, but have contaminated it with NaCl (pseudotensors) from the outset. Classical pseudotensors cannot prove/disprove anything. Nobody knows how they work, once people manage to shape them the way they want them ("Never make a calculation until you know roughly what the answer will be!", John A. Wheeler), in order to calculate the result they know from the outset. Forget it. We need to find 'the right question' in the first place. The puzzle is best explained in MTW, p. 467.

The null energy condition [Ref. 1, p. 5], or rather the averaged null energy condition (ANEC), requires that the null-null component of the stress-energy tensor, integrated along a complete null geodesic, is non-negative for all states (Ulvi Yurtsever, arXiv:gr-qc/9411056v1).

But how do you envisage 'a complete null geodesic' in an "expanding" space? I can't. It is also totally unclear to me how some 'perfectly smooth ideal fluid' (cf. Matt Visser) would couple to type I matter fields, to prove that ANEC is satisfied at all. Surely Mother Nature has made it strictly satisfied, or else we would have observed anomalous, if not catastrophic, events (more above).

But again, these are just my scattered thoughts about the adverb 'marginally'. I think we shouldn't have jumped into conclusion that the "dark puzzle" originates directly from some mundane stuff with positive energy density: it comes out far too much, and suspiciously well-tuned during all cosmological stages. Clearly, there is "a long long way to go" [Ref. 1].

The review by Miao Li et al. [Ref. 1] is a gem. As Alfred North Whitehead noticed, "It requires a very unusual mind to undertake the analysis of the obvious."

p. 153: "However, the problem of understanding the nature of dark energy is as daunting as ever, or perhaps some already hold the key to this understanding without being commonly accepted yet. Clearly, there is a long long way to go for both theorists and experimentalists."

D.C.----Wolfgang Rindler, Relativity, 2nd ed., Oxford University Press, 2006, Sec. 1.14, p. 22: "The equality of inertial and active gravitational mass then remains as puzzling as ever. It would be nice if the inertial mass of an accelerating particle were simply a back-reaction to its own gravitational field, but that is not the case."

I mean, a ' fundamental volume' may be chosen in such way that the "number" of such volumes, in any finite volume of space, to be uncountably infinite (cf. Kurt Gödel, 'What is Cantor's Continuum Problem?', American Mathematical Monthly, Vol. 54, No. 9 (November 1947), pp. 515-525). Which could be interpreted as a non-Archimedean, and purely geometrical "cutoff", at which Alice could see 'the grin of the cat without the cat', placed in the l.h.s. of Einstein equation.

You believe that "there is no reservoir of degrees of freedom available for creating new degrees of freedom as the universe expands", but if the evolution of the universe is non-unitary, such "reservoir" will be needed to account for the *emergence* of new degrees of freedom from 'the unknown unknown'.

Regarding the "interacting theory where the general form of the behavior of the vacuum energy density has the same form as in the free case", and the “Why now?” paradox, please see Sec. Summary, pp. 35-36, in

http://www.god-does-not-play-dice.net/ExplanatoryNote.pdfYour idea that the scale factor (or the lattice spacing) may be changing "in a spatially uniform manner" is one of the toughest issues I've ever faced. I am trying to propose a 'scale relativity principle', which would treat the basic characteristics of 3-D space in a relational manner, that is, the notions of Large vs Small, inside vs outside, and left vs right, will no linger be absolute faculties of 3-D space, but will be relational, that is, 'relative to two observers', A and B. Keep observer A fixed at the length scale of tables and chairs, and allow a table of length 1 m, relative to A, to shrink its "length" toward the Small, while being closely watched by its "co-moving" observer B. As the size of the (relative to A) table and observer B shrink toward the Planck scale, the observer B will always, at all times, measure "the same" length of 1 m of their table, due to altering the metric of space/the scale factor toward the Small. This could only be possible if the "number" of "fundamental volumes" is uncountably infinite. Conversely, if we let the length of the table and observer B to "enlarge" toward the Large, relative to the fixed observer A, the length of the table would look like approaching the cosmological horizon to observer A, while *at the same time* it will keep its relative length of 1 m to its co-moving observer B.

Sorry for this unsolicited, and too long, email. Maybe you have similar ideas, in which case I will be more than happy to study them in details. I just don't want to invent the wheel.

All the best,

Dimi Chakalov-------

Note: I wish to stress that Marvin Weinstein's arXiv:1101.2177v1 [astro-ph.CO] is a must to read and study. It is indeed a fundamental research. The only point at which I disagree with him is this (p. 7): "Clearly my discussion of this question is woefully inadequate and a more serious analysis of these issues within the context of specific models is required in order to better assess the viability of the idea." In my opinion, his discussion isn't "woefully inadequate", but insightful, precise, clear, and professional. Surely we need further analysis, because nobody knows what 'space' is. All I wanted to suggest is to get rid of all fixed qualities of 'space', such as the spatial relations Small vs Large, etc., with the so-called
relative scale principle (RSP). It is very easy to say it, but immensely difficult to actually do it. I hope to offer more on November 25, 2015.

I read your very interesting arXiv:1101.3619v1, and wonder if you can help me find the proper mathematical expression of an old joke about space inversion, 'how to catch a lion'.

If you ask a mathematician how to catch a lion in Sahara, she would probably suggest that, given the existence of at least one lion there, she would drag a cage for lions in the middle of the desert, lock herself up, and then perform space inversion w.r.t. the cage surface, such that all points outside it will be converted inside the cage, and vice versa. At the end of the day, she will find herself outside the cage, while the poor lion will be locked inside. :-)

Question is, would she see a parity-reversal state of the lion, like inverting a right-hand rubber glove into a left-hand one? And how can we mathematically describe such space inversion?

Please note that the intrinsic properties of our physical space are encoded in the relations of left vs right, and inside vs outside or large vs small. Unlike time-reversal symmetry, these transformations do not produce physically indistinguishable outcomes. I believe this asymmetry is the crux of 'space'.

Kindest regards,

Dimi Chakalov----

Note: Back in October 1998, I wrote an email to Chris Isham, in which I offered the idea that the obvious asymmetry of time can only come from asymmetry of space. The reason is that the "separation" of time from space "takes place" only in our twisted imagination, so if we wish to search for some fundamental asymmetry, we should look closely at 'space'. Many years later, I came up with the Arrow of Space. Not only it requires mathematical presentation of 'space inversion', but new mathematical ideas for describing 'potential reality' as well.

NB: Think of the 'cage surface' above as a spherical trapped 2-D surface, which represents the structure of space at macroscopic scale, namely, the length scale at which we can imagine threeprecisely separated entities: a collection of points located inside the cage, another collection of points from the cage surface, and a third collection of points from the space located outside the cage surface. This is also an operational definition of 'finite space', and a criterion about the "points" in such 'finite space' being FAPP 'strictly local'. In the case of quantum, gravitational, and living system, the "points" are quasi-local.

Of course, we first need to find out the mechanism by which only one "charge" of mass is produced in the local mode of spacetime. Is 'potential reality' charge-neutral? How can AVCs (alien visiting craft) fly quietly and unconstrained by inertia-related dynamics (watch William Pawelec, 0:57:00-0:59:20)? Which begs the question of how to "separate" gravitational from inertial mass. Locally, they do look identical in some 'free falling elevator', but this 'elevator' belongs to the local mode of spacetime only. It can't fly like an AVC. More on 25.11.2015.

In your review of the textbook by Hobson, Efstathiou, and Lasenby on amazon.com, you reiterated that "there are no solutions in general relativity with point particle stress-energy (see the paper by geroch and traschen)", and concluded:

"It is fine to present the calculation that point particle stress-energy will be conserved only for geodesic motion, but don't pretend there is anything more to it than a (very) suggestive calculation. Since no solutions exist for that stress-energy, you haven't shown anything about the motion of particles in GR. (At thevery least, don't discuss the field equations without pointing out that there are no solutions!)"

Surely you and Bob Wald discussed the field equations *and* pointed out that there are no solutions, but how should gravitational self-force be rigorously defined?

I think “point particles” with gravitational self-force do not make sense in linearized approximations of GR. Distributional solutions of Einstein’s equation with support on a timelike hypersurface (“shells”) do not make any sense whatsoever, because they require 'spherical cows' -- linearized Einstein equation and linearized Bianchi identity.

With such 'spherical cows' you and Bob are eliminating the non-linear effect you are supposed to describe:

"... gravitational radiation involves spacetimes that are highly non-symmetrical and highly dynamical. No exact solution of Einstein's equations is known that describes the emission and propagation of gravitational waves from a source, and the reaction of the source to the emission of those waves.

"As a result, most of our understanding of gravitational radiation has come from approximations to Einstein's equations. (...) The underlying idea is to treat space-time as being that of flat Minkowski spacetime as the zeroth approximation, and to modify it by successive corrections.....

"Nevertheless it is no less mysterious: we have no good understanding of why this approximation (post-post-linear approximation - D.C.) to general relativity should be so effective.....

"However, the discovery revealed an ugly truth about the "problem of motion". As Ehlers et al. pointed out in an influential 1976 paper [17], the general relativistic problem of motion and radiation was full of holes large enough to drive trucks through. They pointed out that most treatments of the problem used "delta functions" as a way to approximate the bodies in the system as point masses.

"As a consequence, the "self-field", the gravitational field of the body evaluated at its own location, becomes infinite. While this is not a major issue in Newtonian gravity or classical electrodynamics, the non-linear nature of general relativity requires that this infinite self-field contribute to gravity. In the past, such infinities had been simply swept under the rug.

"Similarly, because gravitational energy itself produces gravity it thus acts as a source throughout spacetime. This means that, when calculating radiative fields, integrals for the multipole moments of the source that are so useful in treating radiation begin to diverge.

"These divergent integrals had also been routinely swept under the rug. Ehlers et al. further pointed out that the true boundary condition for any problem involving radiation by an isolated system should be one of "no incoming radiation" from the past. Connecting this boundary condition with the routine use of retarded solutions of wave equations was not a trivial matter in general relativity. Finally they pointed out that there was no evidence that the post-Newtonian approximation, so central to the problem of motion, was a convergent or even asymptotic sequence. Nor had the approximation been carried out to high enough order to make credible error estimates.....

"The one question that remains open is the nature of the post-Newtonian sequence; we still do not know if it converges, diverges or is asymptotic.....

"Wigner remarked that the effectiveness of mathematics in the natural sciences was mysterious. The unreasonable effectiveness of the post-Newtonian approximation in gravitational physics is no less mysterious."

I haven't been able to understand the Equivalence Principle and its various formulations, so I very much applaud your suggestion that gravitational energy momentum p_G is different by its very nature from the inertial energy-momentum p_I.

Yet it seems to me that their observed numerical equality is not accidental, since any residual energy-momentum from numerically different values of p_G and p_I would look like coming from some "ether". That is, in the framework of present-day GR, such residual energy-momentum would look like "dark energy of the ether". Which may be the case chosen by Mother Nature:

I am surprised that you again omitted Henry Margenau in your recent paper [Ref. 1]. I looked in your latest book "On Physics and Philosophy" (Princeton University Press, 2006; ISBN: 978-0691119649), and found out that John Bell was cited 51 times, yet you haven't mentioned Henry Margenau, not even once. Which is rather odd, given the fact that his views were published many years prior to yours: